Chapter 7, Estimation and Data Analysis Video Solutions, Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Problem 1

Suppose the random variable $X$ has the density function
$$
f(x)= \begin{cases}(1+\lambda) x^\lambda, & 0<x<1 \\ 0 & \text { otherwise. }\end{cases}
$$
Show that the maximum likelihood estimate of $\lambda$ based on a given random sample of size $n$ is given by
$$
\hat{\lambda}=-\left(1+\frac{n}{\sum_{i=1}^n \ln x_i}\right) .
$$

Check back soon!

00:57

Problem 2

Consider a uniform random variable defined on the interval $0 \leq$ $x \leq \beta$, where $\beta$ is unknown. Use the method of moments to find an estimate of $\beta$ based on a given random sample of size $n$.

Manik Pulyani

Numerade Educator

02:31

Problem 3

If the $n=20$ values of processing time mentioned in Example 7.1 .2 are $16.39,25.09,16.31,20.94,17.58,19.06,17.21,18.48,16.88$, $15.51,25.87,17.63,29.13,21.34,11.14,26.03,23.28,21.13,18.46$, 14.25 , find the method of moments estimates of $\beta$ and $\alpha$.

Aimal Hassan

Numerade Educator

01:07

Problem 4

If the $n=20$ values of number of customer arrivals in a 10 minute period mentioned in Example 7.1.3 are 25, 32, 34, 22, 27, 29, $23,22,30,31,33,21,28,25,24,35,27,30,34,26$, find the method of moments estimates of $\alpha$ and $\sigma$.

Carson Merrill

Numerade Educator

03:34

Problem 5

Bortkiewicz [5] in 1898 fitted the Poisson distribution to the number of deaths from horse kicks in the Prussian cavalry per corps-year for each of 200 corps-years. His data is given in the table. Estimate the Poisson parameter $\alpha$ using the method of maximum likelihood and construct a $95 \%$ confidence interval for it using $\hat{\sigma}=\sqrt{\hat{\alpha}}$.
$$
\begin{array}{|c|c|}
\hline \text { Number of Deaths } & \begin{array}{l}
\text { Observed Number } \\
\text { Corps-Years }
\end{array} \\
\hline 0 & 109 \\
\hline 1 & 65 \\
\hline 2 & 22 \\
\hline 3 & 3 \\
\hline 4 & 1 \\
\hline \geq 5 & 0 \\
\hline \text { Total } & 200 \\
\hline
\end{array}
$$

Kari Hasz

Numerade Educator

05:11

Problem 6

Consider Example 7.1.3.
(a) Prove that
$$
\sum_{i=1}^n\left(X_i-\bar{X}\right)^2=\sum_{i=1}^n X_i^2-n(\bar{X})^2 .
$$
(b) Using (a) show that
$$
M_2-M_1^2=\left(\frac{n-1}{n}\right) S^2 .
$$

Anurag Kumar

Numerade Educator

05:18

Problem 7

Consider the ratio (7.37) in Example 7.1.8. Show that this ratio is larger than unity (that is, that $p_k(N)>p_k(N-1)$ ) if and only if $N k<r n$.

Arjun Singh

Numerade Educator

00:58

Problem 8

The farm experts at Fanny Farmers Farm plan to lay out a square plot of land with side $\mu$ by using a long rod of length $\mu$ so the area of the plot is $\mu^2$. Unfortunately, the length of the rod is not known exactly, so $n$ independent measurements by $n$ independent farmers are taken, yielding the values $x_1, x_2, \ldots, x_n$. We assume that each $X_i$ has mean $\mu$ and variance $\sigma^2$.
(a) Show that $(\bar{X})^2$ is not an unbiased estimator of $\mu^2$, the area of the field.
(b) For what value of $k$ is the estimator $(\bar{X})^2-k \times S^2$, where $S^2$ is the sample variance, an unbiased estimator for $\mu^2$ ?

Clarissa Noh

Numerade Educator

06:32

Problem 9

Consider the Pascal distribution described in Chapter 3 preceding Exercise 20. It describes a sequence of Bernoulli trials, that continues until $r$ successes occur. The probability that $k$ failures occur before the $r$ th success is given by
$$
p(k ; r, p)=\left(\begin{array}{c}
r+k-1 \\
k
\end{array}\right) \times p^k \times q^k,
$$
for $k=0,1, \ldots$, where $q=1-p$.
(a) Suppose $r \geq 2$. To estimate $p$, suppose trials are continued until $r$ successes are achieved. Suppose $k$ failures occur before the $r$ th success. Then let $\hat{p}=(r-1) /(k+r-1)$. Show that $\hat{p}$ is an unbiased estimator of $p$.
(b) The Vice President of Information Systems wants to interview five Information Center users who are satisfied with the service. An assistant polls a number of users. The sequence of responses is SSFFSFFFSS, where S means the user is satisfied and F that he or she is not. Estimate the true proportion of Information Center users who are satisfied.

Robin Corrigan

Numerade Educator

03:37

Problem 10

Elvin Prellvin, the chief statistician for Heartbreak Hotels, is provided with a random sample from a normal population with $\bar{x}=$ 12.9 and $s=3.2$.
(a) If $n=121$, what should Elvin obtain as a $95 \%$ confidence interval for $\mu$ and for $\sigma^2$ ?
(b) Answer (a), if $n=9$ rather than 121 .

Jeremiah Mbaria

Numerade Educator

01:26

Problem 11

Renaissance Resistors made a random check of 50 of their professional employees who had been provided with either a personal computer or a computer workstation and found that 22 of them used their computers for more than 6 hours per day.
(a) Find the maximum likelihood estimate of the true proportion of professionals who use their computers at least 6 hours per day.
(b) Find the $95 \%$ confidence interval for the estimate of part (a).
(c) How large should the random sample be to ensure that the width of the $95 \%$ confidence interval of part (b) does not exceed 0.1 ?