• Home
  • Textbooks
  • Probability, Statistics, and Random Processes For Electrical Engineering
  • Statistic

Probability, Statistics, and Random Processes For Electrical Engineering

Alberto Leon-Garcia

Chapter 8

Statistic - all with Video Answers

Educators

Chapter Questions

10:10

Problem 1

Let $X$ be a Gaussian random variable with mean 10 and variance 4. A sample of size 9 is obtained and the sample mean, minimum, and maximum of the sample are calculated.
(a) Find the probability that the sample mean is less than 9.
(b) Find the probability that the minimum is greater than 8.
(c) Find the probability that the maximum is less than 12.
(d) Find $n$ so the sample mean is within 1 of the true mean with probability 0.95 .
(e) Generate 100 random samples of size 9. Compare the probabilities obtained in parts $a, b$, and $c$ to the observed relative frequencies.

Sherrie Fenner
Sherrie Fenner
Numerade Educator
View

Problem 2

The lifetime of a device is an exponential random variable with mean 50 months. A sample of size 25 is obtained and the sample mean, maximum, and minimum of the sample are calculated.
(a) Estimate the probability that the sample mean differs from the true mean by more than 1 month.
(b) Find the probability that the longest-lived sample is greater than 100 months.
(c) Find the probability that the shortest-lived sample is less than 25 months.
(d) Find $n$ so the sample mean is within 5 months of the true mean with probability 0.9 .
(e) Generate 100 random samples of size 25 . Compare the probabilities obtained in parts $a, b$, and $c$ to the observed relative frequencies.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 3

Let the signal $X$ be a uniform random variable in the interval $[-3,3]$, and suppose that a sample of size 50 is obtained.
(a) Estimate the probability that the sample mean is outside the interval $[-0.5,0.5]$.
(b) Estimate the probability that the maximum of the sample is less than 2.5 .
(c) Estimate the probability that the sample mean of the squares of the samples is greater than 3 .
(d) Generate 100 random samples of size 50. Compare the probabilities obtained in parts $\mathrm{a}, \mathrm{b}$, and c to the observed relative frequencies.

Check back soon!
03:10

Problem 4

Let $X$ be a Poisson random variable with mean $\alpha=2$, and suppose that a sample of size 16 is obtained.
(a) Estimate the probability that the sample mean is greater than 2.5 .
(b) Estimate the probability that the sample mean differs from the true mean by more than 0.5.
(c) Find $n$ so the sample mean differs from the true mean by more than 0.5 with probability 0.95 .
(d) Generate 100 random samples of size 16. Compare the probabilities obtained in parts $a$ and $b$ to the observed relative frequencies.

Amany Waheeb
Amany Waheeb
Numerade Educator
13:28

Problem 5

The interarrival time of queries at a call center are exponential random variables with mean interarrival time $1 / 4$. Suppose that a sample of size 9 is obtained.
(a) The estimator $\hat{\lambda}_1=1 / \bar{X}_9$ is used to estimate the arrival rate. Find the probability that the estimator differs from the true arrival rate by more than 1 .
(b) Suppose the estimator $\hat{\lambda}_2=1 / 9 \min \left(X_1, \ldots, X_9\right)$ is used to estimate the arrival rate. Find the probability that the estimator differs from the true arrival rate by more than 1.
(c) Generate 100 random samples of size 9. Compare the probabilities obtained in parts a and b to the observed relative frequencies.

Saeeda Aman
Saeeda Aman
Numerade Educator
View

Problem 6

Let the sample $X_1, X_2, \ldots, X_n$ consist of iid versions of the random variable $X$. The method of moments involves estimating the moments of $X$ as follows:

$$
\hat{m}_k=\frac{1}{n} \sum_{j=1}^n X_j^k
$$

(a) Suppose that $X$ is a uniform random variable in the interval $[0, \theta]$. Use $\hat{m}_1$ to find an estimator for $\theta$.
(b) Find the mean and variance of the estimator in part a.

Victor Salazar
Victor Salazar
Numerade Educator
01:40

Problem 7

Let $X$ be a gamma random variable with parameters $\alpha$ and $\beta=1 / \lambda$.
(a) Use the first two moment estimators $\hat{m}_1$ and $\hat{m}_2$ of $X$ (defined in Problem 8.6) to estimate the parameters $\alpha$ and $\beta$.
(b) Describe the behavior of the estimators in part a as $n$ becomes large.

Adriano Chikande
Adriano Chikande
Numerade Educator
View

Problem 8

Let $\mathbf{X}=(X, Y)$ be a pair of random variables with known means, $\mu_1$ and $\mu_2$. Consider the following estimator for the covariance of $X$ and $Y$ :

$$
\hat{C}_{X, Y}=\frac{1}{n} \sum_{j=1}^n\left(X_j-\mu_1\right)\left(Y_j-\mu_2\right)
$$

(a) Find the expected value and variance of this estimator.
(b) Explain the behavior of the estimator as $n$ becomes large.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 9

Let $\mathbf{X}=(X, Y)$ be a pair of random variables with unknown means and covariances. Consider the following estimator for the covariance of $X$ and $Y$ :

$$
\hat{K}_{X, Y}=\frac{1}{n-1} \sum_{j=1}^n\left(X_j-\bar{X}_n\right)\left(Y_j-\bar{Y}_n\right) .
$$

(a) Find the expected value of this estimator.
(b) Explain why the estimator approaches the estimator in Problem 8.8 for $n$ large. Hint: See Eq. (8.15).

Victor Salazar
Victor Salazar
Numerade Educator
05:49

Problem 10

Let the sample $X_1, X_2 \ldots, X_n$ consist of iid versions of the random variable $X$. Consider the maximum and minimum statistics for the sample:

$$
W=\min \left(X_1, \ldots, X_n\right) \quad \text { and } \quad Z=\max \left(X_1, \ldots, X_n\right)
$$

(a) Show that the pdf of $Z$ is $f_Z(x)=n\left[F_X(x)\right]^{n-1} f_X(x)$.
(b) Show that the pdf of $W$ is $f_W(x)=n\left[1-F_X(x)\right]^{n-1} f_X(x)$.

Philomena Marfo
Philomena Marfo
Numerade Educator
01:13

Problem 11

Show that the mean square estimation error satisfies $E\left[(\hat{\Theta}-\theta)^2\right]=\operatorname{VAR}[\hat{\Theta}]+B(\hat{\Theta})^2$.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 12

Let the sample $X_1, X_2, X_3, X_4$ consist of iid versions of a Poisson random variable $X$ with mean $\alpha=4$. Find the mean and variance of the following estimators for $\alpha$ and determine whether they are biased or unbiased.
(a) $\hat{\alpha}_1=\left(X_1+X_2\right) / 2$.
(b) $\hat{\alpha}_2=\left(X_3+X_4\right) / 2$.
(c) $\hat{\alpha}_3=\left(X_1+2 X_2\right) / 3$.
(d) $\hat{\alpha}_4=\left(X_1+X_2+X_3+X_4\right) / 4$.

Victor Salazar
Victor Salazar
Numerade Educator
02:44

Problem 13

(a) Let $\hat{\Theta}_1$ and $\hat{\Theta}_2$ be unbiased estimators for the parameter $\theta$. Show that the estimator $\hat{\Theta}=p \hat{\Theta}_1+(1-p) \hat{\Theta}_2$ is also an unbiased estimator for $\theta$, where $0 \leq p \leq 1$.
(b) Find the value of $p$ in part a that minimizes the mean square error.
(c) Find the value of $p$ that minimizes the mean square error if $\hat{\Theta}_1$ and $\hat{\Theta}_2$ are the estimators in Problems 8.12a and 8.12b.
(d) Repeat part c for the estimators in Problems 8.12a and 8.12d.
(e) Let $\hat{\Theta}_1$ and $\hat{\Theta}_2$ be unbiased estimators for the first and second moments of $X$. Find an estimator for the variance of $X$. Is it biased?

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 14

The output of a communication system is $Y=\theta+N$, where $\theta$ is an input signal and $N$ is a noise signal that is uniformly distributed in the interval [0, 2]. Suppose the signal is transmitted $n$ times and that the noise terms are iid random variables.
(a) Show that the sample mean of the outputs is a biased estimator for $\theta$.
(b) Find the mean square error of the estimator.

Victor Salazar
Victor Salazar
Numerade Educator
01:22

Problem 15

The number of requests at a Web server is a Poisson random variable $X$ with mean $\alpha=2$ requests per minute. Suppose that $n 1$-minute intervals are observed and that the number $N_0$ of intervals with zero arrivals is counted. The probability of zero arrivals is then estimated by $\hat{p}_0=N_0 / n$. To estimate the arrival rate $\alpha, \hat{p}$ is set equal to the probability of zero arrivals in one minute:

$$
\hat{p}_0=N_0 / n=P[X=0]=\frac{\alpha^0}{0!} e^{-\alpha}=e^{-\alpha} .
$$

(a) Solve the above equation for $\hat{\alpha}$ to obtain an estimator for the arrival rate.
(b) Show that $\hat{\alpha}$ is biased.
(c) Find the mean square error of $\hat{\alpha}$.
(d) Is $\hat{\alpha}$ a consistent estimator?

Amany Waheeb
Amany Waheeb
Numerade Educator

Problem 16

Generate 100 samples size 20 of the Poisson random variables in Problem 8.15.
(a) Estimate the arrival rate $\alpha$ using the sample mean estimator and the estimator from Problem 8.15.
(b) Compare the bias and mean square error of the two estimators.

Check back soon!
08:59

Problem 17

To estimate the variance of a Bernoulli random variable $X$, we perform $n$ iid trials and count the number of successes $k$ and obtain the estimate $\hat{p}=k / n$. We then estimate the variance of $X$ by

$$
\hat{\sigma}^2=\hat{p}(1-\hat{p})=\frac{k}{n}\left(1-\frac{k}{n}\right) .
$$

(a) Show that $\hat{\sigma}^2$ is a biased estimator for the variance of $X$.
(b) Is $\hat{\sigma}^2$ a consistent estimator for the variance of $X$ ?
(c) Find a constant $c$, so that $c \hat{\sigma}^2$ is an unbiased estimator for the variance of $X$.
(d) Find the mean square errors of the estimators in parts b and c .

Willis James
Willis James
Numerade Educator
View

Problem 18

Let $X_1, X_2, \ldots, X_n$ be a random sample of a uniform random variable that is uniformly distributed in the interval $[0, \theta]$. Consider the following estimator for $\theta$ :

$$
\hat{\Theta}=\max \left\{X_1, X_2, \ldots, X_n\right\}
$$

(a) Find the pdf of $\hat{\Theta}$ using the results of Problem 8.10 .
(b) Show that $\hat{\Theta}$ is a biased estimator.
(c) Find the variance of $\hat{\Theta}$ and determine whether it is a consistent estimator.
(d) Find a constant $c$ so that $c \hat{\Theta}$ is an unbiased estimator.
(e) Generate a random sample of 20 uniform random variables with $\theta=5$. Compare the values provided by the two estimators in 100 separate trials.
(f) Generate 1000 samples of the uniform random variable, updating the estimator value every 50 samples. Can you discern the bias of the estimator?

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 19

Let $X_1, X_2, \ldots, X_n$ be a random sample of a Pareto random variable:

$$
f_X(x)=k \frac{\theta^k}{x^{k+1}} \quad \text { for } \theta \leq x
$$

with $k=2.5$. Consider the estimator for $\theta$ :

$$
\hat{\Theta}=\min \left\{X_1, X_2, \ldots X_n\right\}
$$

(a) Show that $\hat{\Theta}$ is a biased estimator and find the bias.
(b) Find the mean squared error of $\hat{\Theta}$.
(c) Determine whether $\hat{\Theta}$ is a consistent estimator.
(d) Use Octave to generate 1000 samples of the Pareto random variable. Update the estimator value every 50 samples. Can you discern the bias of the estimator?
(e) Repeat part d with $k=1.5$. What changes?

Victor Salazar
Victor Salazar
Numerade Educator

Problem 20

Generate 100 samples of sizes $5,10,20$ of exponential random variables with mean 1 . Compare the histograms of the estimates given by the biased and unbiased estimators for the sample variance.

Check back soon!
08:18

Problem 21

Find the variance of the sample variance estimator in Example 8.8. Hint: Assume $m=0$.

Anas Venkitta
Anas Venkitta
Numerade Educator

Problem 22

Generate 100 samples of size 20 of pairs of zero-mean, unit-variance jointly Gaussian random variables with correlation coefficient $\rho=0.50$. Compare the histograms of the estimates given by the estimators for the sample covariance in Problems 8.8 and 8.9.

Check back soon!
View

Problem 23

Repeat the scenario in Problem 8.22 for the following estimator for the correlation coefficient between two random variables $X$ and $Y$ :

$$
\hat{\rho}_{x, y}=\frac{\sum_{j=1}^n\left(X_j-\bar{X}_n\right)\left(Y_j-\bar{Y}_n\right)}{\sqrt{\sum_{j=1}^n\left(X_j-\bar{X}_n\right)^2 \sum_{j=1}^n\left(Y_j-\bar{Y}_n\right)^2}}.
$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 24

Let $X$ be an exponential random variable with mean $1 / \lambda$.
(a) Find the maximum likelihood estimator $\hat{\Theta}_{M L}$ for $\theta=1 / \lambda$.
(b) Find the maximum likelihood estimator $\hat{\Theta}_{M L}$ for $\theta=\lambda$.
(c) Find the pdfs of the estimators in part a.
(d) Is the estimator in part a unbiased and consistent?
(e) Repeat 20 trials of the following experiment: Generate a sample of 16 observations of the exponential random variable with $\lambda=1 / 2$ and find the values given by the estimators in parts a and $b$. Show a histogram of the values produced by the estimators.

Victor Salazar
Victor Salazar
Numerade Educator
08:11

Problem 25

Let $X=\theta+N$ be the output of a noisy channel where the input is the parameter $\theta$ and $N$ is a zero-mean, unit-variance Gaussian random variable. Suppose that the output is measured $n$ times to obtain the random sample $X_i=\theta+N_i$ for $i=1, \ldots, n$.
(a) Find the maximum likelihood estimator $\hat{\Theta}_{M L}$ for $\theta$.
(b) Find the pdf of $\hat{\Theta}_{M L}$.
(c) Determine whether $\hat{\Theta}_{M L}$ is unbiased and consistent.

Abhirup Pal
Abhirup Pal
Numerade Educator
05:22

Problem 26

Show that the maximum likelihood estimator for a uniform random variable that is distributed in the interval $[0, \theta]$ is $\hat{\Theta}=\max \left\{X_1, X_2, \ldots, X_n\right\}$. Hint: You will need to show that the maximum occurs at an endpoint of the interval of parameter values.

Amany Waheeb
Amany Waheeb
Numerade Educator
View

Problem 27

Let $X$ be a Pareto random variable with parameters $\alpha$ and $x_m$.
(a) Find the maximum likelihood estimator for $\alpha$ assuming $x_m$ is known.
(b) Show that the maximum likelihood estimators for $\alpha$ and $x_m$ are:

$$
\hat{\alpha}_{M L}=n\left[\sum_{j=1}^n \log \left(\frac{X_j}{\hat{x}_{m, M L}}\right)\right]^{-1} \quad \text { and } \quad \hat{x}_{m, M L}=\min \left(X_1, X_2, \ldots, X_n\right) .
$$

(c) Discuss the behavior of the estimators in parts a and b as $n$ becomes large and determine whether they are consistent.
(d) Repeat five trials of the following experiment: Generate a sample of 100 observations of the Pareto random variable with $\alpha=2.5$ and $x_m=1$ and obtain the values given by the estimators in part $b$. Repeat for $\alpha=1.5$ and $x_m=1$, and $\alpha=0.5$ and
$x_m=1$.

Victor Salazar
Victor Salazar
Numerade Educator
10:03

Problem 28

(a) Show that the maximum likelihood estimator for the parameter $\theta=\alpha^2$ of the Rayleigh random variable is

$$
\hat{\alpha}_{M L}^2=\frac{1}{2 n} \sum_{j=1}^n X_j^2 .
$$

* (b) Is the estimator is unbiased?
(c) Repeat 50 trials of the following experiment: Generate a sample of 16 observations of the Rayleigh random variable with $\alpha=2$ and find the values given by the estimator in part a. Show a histogram of the values produced by the estimator.

Amany Waheeb
Amany Waheeb
Numerade Educator
11:20

Problem 29

(a) Show that the maximum likelihood estimator $f(0, a$ of the beta random variable with $b=1$ is

$$
\hat{a}_{M I}=\left[\frac{1}{n} \sum_{j=1}^n \log X_j\right]^{-1} .
$$

(b) Generate a sample of 100 observations of the beta random variable with $b=1$ and $a=0.5$ to obtain the estimate for $a$. Repeat for $a=1, a=2$, and $a=3$.

Willis James
Willis James
Numerade Educator
03:38

Problem 30

Let $X$ be a Weibull random variable with parameters $\alpha$ and $\beta$ (see Eq. 4.102).
(a) Assuming that $\beta$ is known, show that the maximum likelihood estimator for $\theta=\alpha$ is:

$$
\hat{\alpha}_{M I}=\left[\frac{1}{n} \sum_{j=1}^n X_j^\beta\right]^{-1} .
$$
(b) Generate a sample of 100 observations of the Weibull random variable with $\alpha=1$ and $\beta=1$ to obtain the estimate for $\alpha$. Repeat for $\beta=2$ and $\beta=4$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:01

Problem 31

A certain device is known to have an exponential lifetime.
(a) Suppose that $n$ devices are tested for $T$ seconds, and the number of devices that fail within the testing period is counted. Find the maximum likelihood estimator for the mean lifetime of the device. Hint: Use the invariance property.
(b) Repeat ten trials of the following experiment: Generate a sample of 16 observations of the exponential random variable with $\lambda=1 / 10$ and testing period $T=15$. Find the estimates for the mean lifetime using the method in part a and compare these with the estimates provided by Problem 8.24a.

Dominador Tan
Dominador Tan
Numerade Educator
03:47

Problem 32

Let $X$ be a gamma random variable with parameters $\alpha$ and $\lambda$.
(a) Find the maximum likelihood estimator $\hat{\lambda}_{M L}$ for $\lambda$ assuming $\alpha$ is known.
(b) Find the maximum likelihood estimators $\hat{\alpha}_{M L}$ and $\hat{\lambda}_{M L}$ for $\alpha$ and $\lambda$. Assume that the function $\Gamma^{\prime}(\alpha) / \Gamma(\alpha)$ is known.

Clarissa Noh
Clarissa Noh
Numerade Educator

Problem 33

Let $\mathbf{X}=(X, Y)$ be a jointly Gaussian random vector with zero means, unit variances, and unknown correlation coefficient $\rho$. Consider a random sample of $n$ such vectors.
(a) Show that the ML estimator for $\rho$ involves solving a cubic eqation.
(b) Show that Problem 8.23 gives the ML estimator if the mean and variances are unknown.
(c) Repeat 5 trials of the following: Generate a sample of 100 observations of the pairs of zero-mean, unit-variance Gaussian random variables and estimate $\rho$. using parts a and b for the cases: $\rho=0.5, \rho=0.9$, and $\rho=0$.

Check back soon!
View

Problem 34

(Invariance Property.) Let $\hat{\Theta}_{M L}$ be the maximum likelihood estimator for the parameter $\theta$ of $X$. Suppose that we are interested instead in finding the maximum likelihood estimator for $h(\theta)$, which is an invertible function of $\theta$. Explain why this maximum likelihood estimator is given by $h\left(\hat{\Theta}_{M L}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 35

Show that the Fisher information is also given by Eq. (8.36). Assume that the first two partial derivatives of the likelihood function exist and that they are absolutely integrable so that differentiation and integration with respect to $\theta$ can be interchanged.

Check back soon!
View

Problem 36

Show that the following random variables have the given Cramer-Rao lower bound and determine whether the associated maximum likelihood estimator is efficient:
(a) Binomial with parameters $n$ and unknown $p: p(1-p) / n^2$.
(b) Gaussian with known variance $\sigma^2$ and unknown mean: $\sigma^2 / n$.
(c) Gaussian with unknown variance: $2 \sigma^4 / n$. Consider two cases: mean known; mean unknown. Does the standard unbiased estimator for the variance achieve the Cramer-Rao lower bound? Note that $E\left[(X-\mu)^4\right]=3 \sigma^4$.
(d) Gamma with parameters known $\alpha$ and unknown $\beta=1 / \lambda: \beta^2 / n \alpha$.
(e) Poisson with parameter unknown $\alpha: \alpha / n$.

Victor Salazar
Victor Salazar
Numerade Educator
07:38

Problem 37

Let $\hat{\Theta}_{M L}$ be the maximum likelihood estimator for the mean of an exponential random variable. Suppose we estimate the variance of this exponential random variable using the estimator $\hat{\Theta}_{M L}^2$. What is the probability that $\hat{\Theta}_{M L}^2$ is within $5 \%$ of the true value of the variance? Assume that the number of samples is large.

Jacquelinne S. Mejia Sandoval
Jacquelinne S. Mejia Sandoval
Numerade Educator
13:28

Problem 38

Let $\hat{\Theta}_{M L}$ be the maximum likelihood estimator for the mean $\alpha$ of a Poisson random vart able. Suppose we estimate the probability of no arrivals $P[X=0]=e^{-a}$ with the estimata $e^{-\dot{\Theta}}=L$. Find the probability that this estimator is within $10 \%$ of the true value of $P[X=0$ Assume that the number of samples is large.

Saeeda Aman
Saeeda Aman
Numerade Educator
01:04

Problem 39

A voltage measurement consists of the sum of a constant unknown voltage and a Gauss-ian-distributed noise voltage of zero mean and variance $10 \mu V^2$. Thirty independent measurements are made and a sample mean of $100 \mu \mathrm{~V}$ is obtained. Find the corresponding 95\% confidence interval.

Adriano Chikande
Adriano Chikande
Numerade Educator
04:09

Problem 40

Let $X_j$ be a Gaussian random variable with unknown mean $E[X]=\mu$ and variance 1.
(a) Find the width of the $95 \%$ confidence intervals for $\mu$ for $n=4,16,100$.
(b) Repeat for $99 \%$ confidence intervals.

Lucas Finney
Lucas Finney
Numerade Educator
06:17

Problem 41

The lifetime of 225 light bulbs is measured and the sample mean and sample variance are found to be 223 hr and 100 hr , respectively.
(a) Find a $95 \%$ confidence interval for the mean lifetime.
(b) Find a $95 \%$ confidence interval for the variance of the lifetime.

Willis James
Willis James
Numerade Educator
02:15

Problem 42

Let $X$ be a Gaussian random variable with unknown mean and unknown variance. A set of 10 independent measurements of $X$ yields

$$
\sum_{j=1}^{10} X_j=350 \quad \text { and } \quad \sum_{j=1}^{10} X_j^2=12,645
$$

(a) Find a $90 \%$ confidence interval for the mean of $X$.
(b) Find a $90 \%$ confidence interval for the variance of $X$.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
04:46

Problem 43

Let $X$ be a Gaussian random variable with unknown mean and unknown variance. A set of 10 independent measurements of $X$ yields a sample mean of 57.3 and a sample variance of 23.2.
(a) Find the $90 \%, 95 \%$, and $99 \%$ confidence intervals for the mean.
(b) Repeat part a if a set of 20 measurements had yielded the above sample mean and sample variance.
(c) Find the $90 \%, 95 \%$, and $99 \%$ confidence intervals for the variance in parts a and b.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:07

Problem 44

A computer simulation program is used to produce 150 samples of a random variable. The samples are grouped into 15 batches of ten samples each. The batch sample means are listed below:
$$
\begin{array}{rrrrr}
\hline 0.228 & -1.941 & 0.141 & 1.979 & -0.224 \\
0.501 & -5.907 & -1.367 & -1.615 & -1.013 \\
-0.397 & -3.360 & -3.330 & -0.033 & -0.976 \\
\hline
\end{array}
$$
(a) Find the $90 \%$ confidence interval for the sample mean.
(b) Repeat this experiment by generating beta random variables with parameters $\alpha=2$ and $\beta=3$.
(c) Repeat part b using gamma random variables with $\lambda=1$ and $\alpha=2$.
(d) Repeat part b using Pareto random variables with $x_m=1$ and $\alpha=3 ; x_m=1$ and $\alpha=1.5$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
06:22

Problem 45

A coin is flipped a total of 500 times, in 10 batches of 50 flips each. The number of heads in each of the batches is as follows:

$$
24,27,22,24,25,24,28,26,23,26
$$
(a) Find the $95 \%$ confidence interval for the probability of heads $p$ using the method of batch means.
(b) Simulate this experiment by generating Bernoulli random variables with $p=0.25$; $p=0.01$.

Danielle Flores
Danielle Flores
Numerade Educator
01:39

Problem 46

This exercise is intended to check the statement: "If we were to compute confidence intervals a large number of times, we would find that approximately $(1-\alpha) \times 100 \%$ of the time, the computed intervals would contain the true value of the parameter."
(a) Assuming that the mean is unknown and that the variance is known, find the $90 \%$ confidence interval for the mean of a Gaussian random variable with $n=10$.
(b) Generate 500 batches of 10 zero-mean, unit-variance Gaussian random variables. and determine the associated confidence intervals. Find the proportion of confidence intervals that include the true mean (which by design is zero). Is this in agreement with the confidence level $1-\alpha=.90$ ?
(c) Repeat part b using exponential random variables with mean one. Should the proportion of intervals including the true mean be given by $1-\alpha$ ? Explain.

Manik Pulyani
Manik Pulyani
Numerade Educator
04:27

Problem 47

Generate $160 X_i$ that are uniformly distributed in the interval $[-1,1]$.
(a) Suppose that $90 \%$ confidence intervals for the mean are to be produced. Find the confidence intervals for the mean using the following combinations:

4 batches of 40 samples each,
8 batches of 20 samples each,
16 batches of 10 samples each, and
32 batches of 5 samples each.
(b) Redo the experiment in part a 500 times. In each repetition of the experiment, compute the four confidence intervals defined in part a. Calculate the proportion of time in which the above four confidence intervals include the true mean. Which of the above combinations of the batch size and number of batches are in better agreement with the results predicted by the confidence level? Explain why.

Andrew Kim
Andrew Kim
Numerade Educator
02:18

Problem 48

This exercise explores the behavior of confidence intervals as the number of samples is increased. Generate 1000 samples of independent Gaussian random variables with mean 25 and variance 36. Update and plot the confidence intervals for the mean and variance every 50 samples.

Christopher Stanley
Christopher Stanley
Numerade Educator
03:02

Problem 49

A new Web page design is intended to increase the rate at which customers place orders Prior to the new design, the number of orders in an hour was a Poisson random variable with mean 30 . Eight one-hour measurements with the new design find an average of 32 orders completed per hour.
(a) At a 5\% significance level, do the data support the claim that the order placement rate has increased?
(b) Repeat part a at a $1 \%$ significance level.

Neel Faucher
Neel Faucher
Numerade Educator

Problem 50

Carlos and Michael play a game where each flips a coin once: If the outcomes of the tosses are the same, then no one wins; but if the outcome is different the player with "heads" wiss Michael uses a fair coin but he suspects that Carlos is using a biased coin.
(a) Find a $10 \%$ significance level test for an experiment that counts how many times los wins in 6 games to test whether Carlos is cheating. Repeat for $n=12$ games.
(b) Now design a $10 \%$ significance level test based on the number of times Carlof tosses come up heads. Which test is more effective?
(c) Find the probability of detection if Carlos uses a coin with $p=0.75 ; p=0.55$.

Check back soon!
03:41

Problem 51

The output of a receiver is the sum of the input voltage and a Gaussian random variable with zero mean and variance 4 volt $^2$. A scientist suspects that the receiver input is not properly calibrated and has a nonzero input voltage in the absence of a true input signal.
(a) Find a $1 \%$ significance level test involving $n$ independent measurements of the output to test the scientist's hunch.
(b) What is the outcome of the test if 10 measurements yield a sample mean of -0.75 volts?
(c) Find the probability of a Type II error if there is indeed an input voltage of 1 volt; of 10 millivolts.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:03

Problem 52

(a) Explain the relationship between the $p$-value and the level $\alpha$ of a test.
(b) Explain why the $p$-value provides more information about the test statistic than simply stating the outcome of the hypothesis test.
(c) How should the $p$-value be calculated in a one-sided test?
(d) How should the $p$-value be calculated in a two-sided test?

Marc Lauzon
Marc Lauzon
Numerade Educator

Problem 53

The number of photons counted by an optical detector is a Poisson random variable with known mean $\alpha$ in the absence of a target and known mean $\beta=6>\alpha=2$ when a target is present. Let the null hypothesis correspond to "no target present."
(a) Use the Neyman-Pearson method to find a hypothesis test where the false alarm probability is set to $5 \%$.
(b) What is the probability of detection?
(c) Suppose that $n$ independent measurements of the input are taken. Use trial and error to find the value of $n$ required to achieve a false alarm probability of $5 \%$ and a probability of detection of $90 \%$.

Check back soon!
04:47

Problem 54

The breaking strength of plastic bags is a Gaussian random variable. Bags from company 1 have a mean strength of 8 kilograms and a variance of $1 \mathrm{~kg}^2$; bags from company 2 have a mean strength of 9 kilograms and a variance of $1 \mathrm{~kg}^2$. We are interested in determining whether a batch of bags comes from company 1 (null hypothesis). Find a hypothesis test and determine the number of bags that needs to be tested so that $\alpha$ is $1 \%$ and the probability of detection is $99 \%$.

Nick Johnson
Nick Johnson
Numerade Educator
03:28

Problem 55

Light Internet users have session times that are exponentially distributed with mean 2 hours, and heavy Internet users have session times that are exponentially distributed with mean 4 hours.
* (a) Use the Neyman-Pearson method to find a hypothesis test to determine whether a given user is a light user. Design the test for $\alpha=5 \%$.
(b) What is the probability of detecting heavy users?

Gaurav Kalra
Gaurav Kalra
Numerade Educator
View

Problem 56

Normal Internet users have session times that are Pareto distributed with mean 3 hours and $a=3$, and heavy peer-to-peer users have session times that are Pareto distributed with $a=8 / 7$ and mean 16 hours.
(a) Use the Neyman-Pearson method to find a hypothesis test to determine whether a given user is a normal user. Design the test for $\alpha=1 \%$
(b) What is the probability of detecting heavy peer-to-peer users?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
06:22

Problem 57

Coin factories A and B produce coins for which the probability of heads $p$ is a betadistributed random variable. Factory A has parameters $a=b=10$, and factory B has $a=b=5$.
(a) Design a hypothesis test for $\alpha=5 \%$ to determine whether a batch is from factory A .
(b) What is the probability of detecting factory B coins? Hint: Use the Octave function beta_inv. Assume that the probability of heads in the batch can be determined accurately.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:37

Problem 58

When operating correctly (null hypothesis), wires from a production line have a mean diameter of 2 mm , but under a certain fault condition the wires have a mean diameter of 1.75 mm . The diameters are Gaussian distributed with variance $.04 \mathrm{~mm}^2$. A batch of 10 sample wires is selected and the sample mean is found to be 1.82 mm .
(a) Design a test to determine whether the line is operating correctly. Assume a false alarm probability of $5 \%$.
(b) What is the probability of detecting the fault condition?
(c) What is the $p$-value for the above observation?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:50

Problem 59

Coin 1 is fair and coin 2 has probability of heads $3 / 4$. A test involves flipping a coin repeatedly until the first occurrence of heads. The number of tosses is observed.
(a) Can you design a test to determine whether the fair coin is in use? Assume $\alpha=5 \%$. What is the probability of detecting the biased coin?
(b) Repeat part a if the biased coin has probability $1 / 4$.

Amany Waheeb
Amany Waheeb
Numerade Educator
01:31

Problem 60

The output of a radio signal detection system is the sum of an input voltage and a zeromean, unit-variance Gaussian random variable.
(a) Design a hypothesis test, at a level $\alpha=10 \%$, to determine whether there is a nonzero input assuming $n$ independent measurements of the receiver output (so the additive noise terms are iid random variables).
(b) Find expressions for the Type II error probability and the power of the test in part a.
(c) Plot the power of the test in part a as the input voltage varies from $-\infty$ to $+\infty$ for $n=4,16,64,256$.

Victor Salazar
Victor Salazar
Numerade Educator
01:41

Problem 61

(a) In Problem 8.60, design a hypothesis test, at a level $\alpha$, to determine whether there is a positive input assuming $n$ independent measurements.
(b) Find expressions for the Type II error probability and the power of the test in part a.
(c) Plot the power of the test in part a as the input voltage varies from $-\infty$ to $+\infty$ for $n=4,16,64,256$.

Tyler Moulton
Tyler Moulton
Numerade Educator
00:52

Problem 62

Compare the power curves obtained in Problems 8.60 and 8.61 . Explain why the test in Problem 8.61 is uniformly most powerful, while the test in Problem 8.60 is not.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:50

Problem 63

Consider Example 8.27 where we considered

$$
\begin{aligned}
& H_0: X \text { is Gaussian with } \mu=0 \text { and } \sigma_X^2=1 \\
& H_1: X \text { is Gaussian with } \mu>0 \text { and } \sigma_X^2=1
\end{aligned}
$$

Let $n=25, \alpha=5 \%$. For $\mu=k / 2, k=0,1,2, \ldots, 5$ perform the following experiment: Generate 500 batches of size 25 of the Gaussian random variable with mean $\mu$ and unit variance. For each batch determine whether the hypothesis test accepts or rejects the null hypothesis. Count the number of Type I errors and Type II errors. Plot the empirically obtained power function as a function of $\mu$.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 64

Repeat Problem 8.63 for the following hypothesis test:

$$
\begin{aligned}
& H_0: X \text { is Gaussian with } \mu=0 \text { and } \sigma_X^2=1 \\
& H_1: X \text { is Gaussian with } \mu \neq 0 \text { and } \sigma_X^2=1
\end{aligned}
$$

Let $n=25, \alpha=5 \%$, and run the experiments for $\mu= \pm k / 2, k=0,1,2, \ldots, 5$.

Check back soon!
03:08

Problem 65

Consider the following three tests for a fair coin:
(i) $H_0: p=0.5$ vs. $H_1: p \neq 0.5$
(ii) $H_0: p=0.5$ vs. $H_1: p>0.5$
(iii) $H_0: p=0.5$ vs. $H_1: p<0.5$.

Assume $n=100$ coin tosses in each test and that the rejection regions for the above tests are selected for $\alpha=1 \%$.
(a) Find the power curves for the three tests as a function of $p$.
(b) Explain the power curve of the two-sided test in comparison to those of the onesided tests.

Alexander Cheng
Alexander Cheng
Numerade Educator
03:26

Problem 66

(a) Consider hypothesis test (i) of Problem 8.65 with $\alpha=5 \%$. For $p=k / 10, k=1$, $2, \ldots, 9$ perform the following experiment: Generate 500 batches of 100 tosses of a coin with probability of heads $p$. For each batch determine whether the hypothesis test accepts or rejects the null hypothesis. Count the number of Type I errors and Type II errors. Plot the empirically obtained power function as a function of $\mu$.
(b) Repeat part a for hypothesis test (ii) of Problem 8.65.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
00:52

Problem 67

Consider the hypothesis test developed in Example 8.26 to test $H_0: m=\mu$ vs. $H_1: m>\mu$. Suppose we use this test, that is, the associated rejection and acceptance region, for the following hypothesis testing problem:

$$
\begin{aligned}
& H_0: X \text { is Gaussian with mean } m \leq \mu \text { and known variance } \sigma^2 \\
& H_1: X \text { is Gaussian with mean } m>\mu \text { and known variance } \sigma^2 .
\end{aligned}
$$

Show that the test achieves level $\alpha$ or better. Hint: Consider the power function of the test in Example 8.26.

Manik Pulyani
Manik Pulyani
Numerade Educator
05:12

Problem 68

A machine produces disks with mean thickness 2 mm . To test the machine after undergoing routine maintenance, 10 sample disks are selected and the sample mean of the thickness is found to be 2.2 mm and the sample variance is found to be $0.04 \mathrm{~mm}^2$.
(a) Find a test to determine if the machine is working properly for $\alpha=1 \%$; $\alpha=5 \%$.
(b) Find the $p$-value of the observation.

Amany Waheeb
Amany Waheeb
Numerade Educator
03:14

Problem 69

A manufacturer claims that its new improved tire design increases tire lifetime from $50,000 \mathrm{~km}$ to $55,000 \mathrm{~km}$. A test of 8 tires gives a sample mean lifetime of $52,500 \mathrm{~km}$ and a sample standard deviation of 3000 km .
(a) Find a test to determine if the claim can be supported at a level of $\alpha=1 \%$; $\alpha=5 \%$.
(b) Find the $p$-value of the observation.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
View

Problem 70

A class of 100 engineering freshmen is provided with new laptop computers. The manufacturer claims the charge in the batteries will last four hours. The frosh run a test and find a sample mean of 3.3 hours and a sample standard deviation of 0.5 hours.
(a) Find a test to determine if the manufacturer's claim can be supported at a level of $\alpha=1 \% ; \alpha=5 \%$.
(b) Find the $p$-value of the observation.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:50

Problem 71

Consider the hypothesis test considered in Example 8.29:

$$
\begin{aligned}
& H_0: X \text { is Gaussian with } \mu=0 \text { and } \sigma_X^2 \text { unknown } \\
& H_1: X \text { is Gaussian with } \mu \neq 0 \text { and } \sigma_X^2 \text { unknown. }
\end{aligned}
$$

Let $n=9, \alpha=5 \%, \sigma_X=1$. For $\mu= \pm k / 2, k=0,1,2, \ldots, 5$ perform the following experiment: Generate 500 batches of size 9 of the Gaussian random variable with mean $\mu$ and unit variance. For each batch determine whether the hypothesis test accepts or rejects the null hypothesis. Count the number of Type I errors and Type II errors. Plot the empirically obtained power function as a function of $\mu$. Compare to the expected results.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 72

Repeat Problem 8.71 for the following hypothesis test:
$H_0: X$ is Gaussian with $\mu=0$ and $\sigma_X^2$ unknown
$H_1: X$ is Gaussian with $\mu>0$ and $\sigma_X^2$ unknown.
Let $n=9, \alpha=5 \%, \sigma_X=1$, and $\mu=k / 2, k=0,1,2, \ldots, 5$.

Check back soon!
View

Problem 73

Consider using the hypothesis test in Example 8.29 when the random variable is not Gaussian. Design tests for $\alpha=5 \%, n=9$ and for $n=25$. For $\mu= \pm k / 2, k=0,1,2, \ldots .5$ perform the following experiment: Let $X$ be a uniform random variable in the interval $[-1 / 2,1 / 2]$. Generate 500 batches of size $n$ of the uniform random variable with mean $\mu$. For each batch determine whether the hypothesis test accepts or rejects the null hypothesis. Count the number of Type I errors and Type II errors. Plot the empirically obtained power function as a function of $\mu$. Compare the empirical data to the values expected for the Gaussian random variable.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 74

Consider using the hypothesis test in Problem 8.73 when the random variable is an exponential random variable. Design tests for $\alpha=5 \%, \mu=1, n=9$ and for $n=25$. Repeat the experiment for $\mu=k / 2, k=1,2, \ldots, 5$. Compare the empirical data to the values expected for the Gaussian random variable.

Shu Naito
Shu Naito
Numerade Educator
03:41

Problem 75

A stealth alarm system works by sending noise signals: A "situation normal" signal is sent by transmitting voltages that are Gaussian iid random variables with mean zero and variance 4; an "alarm" signal is sent by transmitting iid Gaussian voltages with mean zero and variance less than 4.
(a) Find a $1 \%$ level hypothesis test to determine whether the situation is normal (null hypothesis) based on the calculation of the sample variance from $n$ voltage samples.
(b) Find the power of the hypothesis test for $n=8,64,256$ as the variance of the alarm signal is varied.

Amany Waheeb
Amany Waheeb
Numerade Educator
00:56

Problem 76

Repeat Problem 8.75 if the alarm signal uses iid Gaussian voltages that have variance greater than 4.

Chai Santi
Chai Santi
Numerade Educator
02:11

Problem 77

A stealth system summons Agent 00111 by sending a sequence of 71 Gaussian iid random variables with mean zero and variance $\mu_0=7$. Find a hypothesis test (to be implemented in Agent's 00111 wristwatch) to determine, at a $1 \%$ level, that she is being summoned. Plot the probability of Type II error.

Ameer Said
Ameer Said
Numerade Educator
02:31

Problem 78

Consider the hypothesis test in Example 8.30 for testing the variance:
$H_0: X$ is Gaussian with $\sigma_X^2=1$ and $m$ unknown
$H_1: X$ is Gaussian with $\sigma_X^2 \neq 1$ and $m$ unknown.
Let $n=16, \alpha=5 \%, \mu=0$. For $\sigma_X^2=k / 3, k=1,2, \ldots, 6$ perform the following experiment: Generate 500 batches of size 16 of the Gaussian random variable with zero mean and variance $\sigma_X^2$. For each batch determine whether the hypothesis test accepts or rejects the null hypothesis. Count the number of Type I errors and Type II errors. Plot the power function as a function of $\mu$. Compare to the expected results.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:31

Problem 79

Consider using the hypothesis test in Problem 8.78 when the random variable is a uniform random variable. Repeat the experiment where $X$ is now a uniform random variable in the interval $[-1 / 2,1 / 2]$. Compare the empirical data to the values expected fort the Gaussian random variable. Repeat the experiment for $n=9$ and $n=36$.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:23

Problem 80

In this exercise we explore the relation between confidence intervals and hypothesis tes ing. Consider the hypothesis test in Example 8.28 but with a level of $\alpha=5 \%$.
(a) Run 200 trials of the following experiment: Generate 10 samples of $X$ given that $H_0$ is true; determine the confidence interval; determine if the interval includes 0 ; determine if the null hypothesis is accepted.
(b) Is the relative frequency of Type I error as expected?

Maxime Rossetti
Maxime Rossetti
Numerade Educator
01:36

Problem 81

The Premium Pen Factory tests one pen in each batch of 100 pens. The ink-filling machine is bipolar, so pens can write continuously for an exponential duration of mean either $1 / 2$ hour or 5 hours. The machine is in the short-life production mode $10 \%$ of the time. A batch of short-life pens sold as long-life pens results in a loss of $$\$ 5$$, while a batch of long-life pens mistakenly sold as short-life results in a loss of $$\$ 3$$. Find the Bayes decision rule to decide whether a batch is long-life or short-life based on the measured lifetime of the test pen.

James Kiss
James Kiss
Numerade Educator

Problem 82

Suppose we send binary information over an erasure channel. If the input to the channel is " 0 ", then the output is equally likely to be " 0 " or "e" for "erased"; and if the input is " 1 " then the outputs are equally likely to be " 1 " or "e." Assume that $P[\Theta=1]=1 / 4$ $=1-P[\Theta=0]$. and that the cost functions are: $C_{00}=C_{11}=0$ and $C_{01}=b C_{161}$.
(a) For $b=1 / 6,1$, and 6 , find the maximum likelihood decision rule, which picks the input that maximizes the likelihood probability for the observed output. Find the average cost for each case.
(b) For the three cases in part a, find the Bayes decision rule that minimizes the average cost. Find the average cost for each case.

Check back soon!

Problem 83

For the channel in Problem 8.82, suppose we transmit each input twice. The receiver makes its decision based on the observed pair of outputs. Find and compare the maximum likelihood and the Bayes' decision rules.

Check back soon!

Problem 84

When Bob throws a dart the coordinates of the landing point are a Gaussian pair of independent random variables $(X, Y)$ with zero mean and variance 1 . When Rick throws the dart the coordinates are also a Gaussian independent pair but with zero mean and variance 4. Bob and Rick are asked to draw a circle centered about the origin with the inner disk assigned to Bob and the outer ring assigned to Rick.
(a) Whenever either player lands on the other player's area, he must pay a $$\$ 1$$ to the house. Find the disk radius that minimizes the players' average cost.
(b) Repeat part a if Bob must pay $$\$ 2$$ when he lands in Rick's area.

Check back soon!

Problem 85

A binary communications system accepts $\Theta$, which is " 0 " or " 1 ", as input and outputs $X$, " 0 " or " 1 ", with probability of error $P[\Theta \neq X]=p=10^{-3}$. Suppose the sender uses a repetition code whereby each " 0 " or " 1 " is transmitted $n$ independent times, and the receiver makes its decision based on the $n=8$ corresponding outputs. Assume that $1 / 5=P[\Theta=1]=\alpha=1-P[\Theta=0]$.
(a) Find the maximum likelihood decision rule that selects the input which is more likely for the given $n$ outputs. Find the probability of Type I and Type II errors, as well as the overall probability of error $P_c$.
(b) Find the Bayes decision rule that minimizes the probability of error. Find the probability of Type 1 and Type II errors, as well as $P_e$.
(c) For the decision rules in parts a and b find $n$ so that $P_e=10^{-9}$.

Check back soon!

Problem 86

A binary communications system accepts $\Theta$, which is " +1 " or " -1 ", as input and outputs $X=\Theta+N$, where $N$ is a zero-mean Gaussian random variable with variance $\sigma^2$. The sender uses a repetition code where each " +1 " or " -1 " is transmitted $n$ times, and the receiver makes its decision based on the $n$ outputs. Assume $P[\Theta=1]=\alpha=1-P[\Theta=0]$.
(a) Find the maximum likelihood decision rule and evaluate its Type I and Type II error probabilities as well as its overall probability of error.
(b) Find the Bayes decision rule and compare its error probabilities to part a.
(c) Suppose $\sigma$ is such that $P[N>1]=10^{-3}$. Find the value of $n$ in part b, so that $P_e=10^{-9}$.

Check back soon!

Problem 87

A widely used digital radio system transmits pairs of bits at a time. The input to the system is a pair $\left(\Theta_1, \Theta_2\right)$ where $\Theta_i$ can be +1 or -1 and where the output of the channel is a pair of independent Gaussian random variables $(X, Y)$ with variance $\sigma^2$ and means $\Theta_1$ and $\Theta_2$, respectively. Assume $P\left[\Theta_i=1\right]=\alpha=1-P\left[\Theta_i=0\right]$ and that the input bits are independent of each other. The receiver observes the pair $(X, Y)$ and based on their values decides on the input pair $\left(\Theta_1, \Theta_2\right)$.
(a) P!ot $f_{X, Y}\left(x, y \mid \Theta_1, \Theta_2\right)$ for the four possible input pairs.
(b) Let the cost be zero if the receiver correctly identifies the input pair, and let the cost be one otherwise. Show that the Bayes' decision rule selects the input pair $\left(\theta_1, \theta_2\right)$ that maximizes:

$$
f_{X, Y}\left(x, y \mid \theta_1, \theta_2\right) P\left[\Theta_1,=\theta_1, \Theta_2=\theta_2\right] .
$$

(c) Find the four decision regions in the plane when the inputs are equally likely. Show that this corresponds to the maximum likelihood decision rule.

Check back soon!
02:40

Problem 88

Show that the Bayes estimator for the cost function $C(g(\mathbf{X}), \Theta)=|g(\mathbf{X})-\Theta|$, is given by the median of the a posteriori pdf $f_{\Theta}(\theta \mid \mathbf{X})$. Hint: Write the integral for the average cost as the sum of two integrals over the regions $g(\mathbf{X})>\theta$ and $g(\mathbf{X})<\theta$, and then differentiate with respect to $g(\mathbf{X})$.

Amany Waheeb
Amany Waheeb
Numerade Educator

Problem 89

Show that the Bayes' estimator for the cost function in Eq. (8.96) is given by the MAP estimator for $\theta$.

Check back soon!
08:11

Problem 90

Let the observations $X_1, X_2, \ldots, X_n$ be iid Gaussian random variables with unit variance and unknown mean $\Theta$. Suppose that $\Theta$ is itself a Gaussian random variable with mean 0 and variance $\sigma^2$. Find the following estimators:
(a) The minimum mean square estimator for $\Theta$.
(b) The minimum mean absolute error estimator for $\Theta$.
(c) The MAP estimator for $\Theta$.

Abhirup Pal
Abhirup Pal
Numerade Educator
03:52

Problem 91

Let $X$ be a uniform random variable in the interval $(0, \Theta)$, where $\Theta$ has a gamma distribution $f_{\mathrm{Q}}(\theta)=\theta e^{-\theta}$ for $\theta>0$.
(a) Find the estimator that minimizes the mean absolute error.
(b) Find the estimator that minimizes the mean square error.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 92

Let $X$ be a binomial random variable with parameters $n$ and $\Theta$. Suppose that $\Theta$ has a beta distribution with parameters $\alpha$ and $\beta$.
(a) Show that $f_{\Theta}(\theta \mid X=k)$ is a beta pdf with parameters $a+k$ and $\beta+n-k$.
(b) Show that the minimum mean square estimator is then $(\alpha+k) /(\alpha+\beta+n)$.

Check back soon!

Problem 93

Let $X$ be a binomial random variable with parameters $n$ and $\Theta$. Suppose that $\Theta$ is uniform in the interval $[0,1]$. Consider the following cost function which emphasizes the errors at the extreme values of $\theta$ :

$$
C(g(X), \theta)=\frac{(\theta-g(X))^2}{\theta(1-\theta)}
$$
Show that the Bayes estimator is given by

$$
g(k)=\frac{\Gamma(n)}{\Gamma(k) \Gamma(n-k)} \frac{k}{n} .
$$

Check back soon!
03:43

Problem 94

The following histogram was obtained by counting the occurrence of the first digits in telephone numbers in one column of a telephone directory:
Table can't copy
Test the goodness of fit of this data to a random variable that is uniformly distributed in the set $\{0,1, \ldots, 9\}$ at a $1 \%$ significance level. Repeat for the set $\{2,3, \ldots, 9\}$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
06:55

Problem 95

A die is tossed 96 times and the number of times each face occurs is counted:
Table can't copy
(a) Test the goodness of fit of the data to the pmf of a fair die at a 5\% significance level.
(b) Run the following experiment 100 times: Generate 50 iid random variables from the discrete $\operatorname{pmf}\{1 / 6,1 / 6,1 / 6,1 / 6,3 / 24,5 / 24\}$. Test the goodness of fit of this data to tosses from a fair die. What is the relative frequency with which the null hypothesis is rejected?
(c) Repeat part b using a sample size of 100 iid random variables.

Jacquelinne S. Mejia Sandoval
Jacquelinne S. Mejia Sandoval
Numerade Educator
02:21

Problem 96

(a) Show that the $D^2$ statistic when $K=2$ is:

$$
D^2=\frac{\left(n_1-n p_1\right)^2}{n p_1\left(1-p_1\right)}=\left[\frac{\left(n_1-n p_1\right)}{\sqrt{n p_1\left(1-p_1\right)}}\right]^2
$$

(b) Explain why $D^2$ approaches a chi-square random variable with 1 degree of freedom as $n$ becomes large.

Karen Song
Karen Song
Numerade Educator
02:31

Problem 97

(a) Repeat the following experiment 500 times: Generate 100 samples of the sum of $X$ of 10 iid uniform randorm variables from the unit interval. Perform a goodness-of-fit test of the random samples of $X$ to the Gaussian random variable with the same mean and variance. What is the relative frequency with which the null hypothesis is rejected at a $5 \%$ level?
(b) Repeat part a for sums of 20 iid uniform random variables.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 98

Repeat Problem 8.97 for the sum of exponential random variables with mean 1 .

Check back soon!
02:44

Problem 99

A computer simulation program gives pairs of numbers $(X, Y)$ that are supposed to be uniformly distributed in the unit square. Use the chi-square test to assess the goodness of fit of the computer output.

Bryan Meares
Bryan Meares
Numerade Educator
00:38

Problem 100

Use the approach in Problem 8.99 to develop a test for the independence between two random variables $X$ and $Y$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:18

Problem 101

You are asked to characterize the behavior of a new binary communications system in which the inputs are $\{0,1\}$ and the outputs are $\{0,1\}$. Design a series of tests to characterize the errors introduced in transmissions using the system. How would you estimate the probability of error $p$ ? How would you determine whether the $p$ is fixed or whether it varies? How would you determine whether errors introduced by the system are independent of each other? How would you determine whether the errors introduced by the system are dependent on the input?

A M
A M
Numerade Educator

Problem 102

You are asked to characterize the behavior of a new binary communications system in which the inputs are $\{0,1\}$ and the outputs assume a continuum of real values. What tests would you change and what tests would you keep from Problem 8.101?

Check back soon!
11:09

Problem 103

Your summer job with the local bus company entails sitting at a busy intersection and recording the bus arrival times for several routes in a table next to their scheduled times. How would you characterize the arrival time behavior of the buses?

Chris Trentman
Chris Trentman
Numerade Educator

Problem 104

Your friend Khash has a summer job with an Internet access provider that involves characterizing the packet transit times to various key sites on the Internet. Your friend has access to some nifty hardware for generating test packets, including GPS systems, to provide accurate timestamps. How would your friend go about characterizing these transit times?

Check back soon!
05:54

Problem 105

Leigh's summer job is with a startup testing a new optical device. Leigh runs a standard test on these devices to determine their failure rates and failure root causes. He looks at the dependence of failures on the supplier, on impurities in the devices, and on different approaches to preparing the devices. How should Leigh go about characterizing failure rate behavior? How should he identify root causes for failures?

Raymond Matshanda
Raymond Matshanda
Numerade Educator