A simulation model of a proposed new computer system for the...

A simulation model of a proposed new computer system for the Hunkydory Boat Company has been constructed. The model provides an estimate of the utilization, $\rho$, of the central processing unit (CPU) by testing one hundred times every millisecond to determine whether or not it is busy and using the formula $$ \rho=P[\mathrm{CPU} \text { is busy }]=\frac{S_n}{n}, $$ where $n$ is the number of samples and $S_n$ the number of times the CPU is busy. How many samples should be made if $\delta=0.005$ and $\epsilon=0.001$ in the formula $$ P\left[\left|\frac{S_n}{n}\right| \geq \delta\right] \leq \epsilon ? $$ Assume $\rho$ is near 0.5 . 49. [HM18] Show that the density function $f_n$ for a Student's $t$ distribution with $n$ degrees of freedom assumes a unique maximum when $x=0$. 50. [HM30] Consider the gamma function defined by $$ \Gamma(t)=\int_0^{\infty} x^{t-1} e^{-x} d x, \quad t>0 . $$ Prove the following: (a) $\Gamma(t+1)=t \Gamma(t)$ for all $t>0$. (Since $\Gamma(1)=1$, this implies that $\Gamma(n+1)=n !, n=1,2, \cdots)$ (b) Show that $\Gamma(t)$ can be written as $$ \Gamma(t)=2^{1-t} \int_0^{\infty} z^{2 t-1} e^{-\frac{1}{2} z^2} d z, $$ for all $t>0$. (c) Using (b), we can write $$ \Gamma\left(\frac{1}{2}\right)=\sqrt{2} \int_0^{\infty} e^{-\frac{1}{2} z^2} d z, $$ and thus, $$ \begin{aligned} {\left[\Gamma\left(\frac{1}{2}\right)\right]^2 } & =2\left\{\int_0^{\infty} e^{-\frac{1}{2} x^2} d x\right\}\left\{\int_0^{\infty} e^{-\frac{1}{2} y^2} d y\right\} \\ & =2 \int_0^{\infty} \int_0^{\infty} e^{-\frac{1}{2}\left(x^2+y^2\right)} d x d y . \end{aligned} $$ Now use polar coordinates to evaluate the double integral and thereby show that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$. 51. [15] Get High Airlines wants to estimate the fraction of smokers $p$ among their passenger population. The airline plans to use sampling with replacement to determine their estimate $\hat{p}$ of $p$. They set up the requirement that $$ P[|\hat{p}-p| \geq 0.005] \leq 0.05 \text {. } $$ (a) How large a sample should Get High take if nothing is assumed about the size of $p$ ? Assume $\rho$ is near 0.5 .

A simulation model of a proposed new computer system for the Hunkydory Boat Company has been constructed. The model provides an estimate of the utilization, $\rho$, of the central processing unit (CPU) by testing one hundred times every millisecond to determine whether or not it is busy and using the formula
$$
\rho=P[\mathrm{CPU} \text { is busy }]=\frac{S_n}{n},
$$
where $n$ is the number of samples and $S_n$ the number of times the CPU is busy. How many samples should be made if $\delta=0.005$ and $\epsilon=0.001$ in the formula
$$
P\left[\left|\frac{S_n}{n}\right| \geq \delta\right] \leq \epsilon ?
$$
Assume $\rho$ is near 0.5 .
49. [HM18] Show that the density function $f_n$ for a Student's $t$ distribution with $n$ degrees of freedom assumes a unique maximum when $x=0$.
50. [HM30] Consider the gamma function defined by
$$
\Gamma(t)=\int_0^{\infty} x^{t-1} e^{-x} d x, \quad t>0 .
$$
Prove the following:
(a) $\Gamma(t+1)=t \Gamma(t)$ for all $t>0$. (Since $\Gamma(1)=1$, this implies that $\Gamma(n+1)=n !, n=1,2, \cdots)$
(b) Show that $\Gamma(t)$ can be written as
$$
\Gamma(t)=2^{1-t} \int_0^{\infty} z^{2 t-1} e^{-\frac{1}{2} z^2} d z,
$$
for all $t>0$.
(c) Using (b), we can write
$$
\Gamma\left(\frac{1}{2}\right)=\sqrt{2} \int_0^{\infty} e^{-\frac{1}{2} z^2} d z,
$$
and thus,
$$
\begin{aligned}
{\left[\Gamma\left(\frac{1}{2}\right)\right]^2 } & =2\left\{\int_0^{\infty} e^{-\frac{1}{2} x^2} d x\right\}\left\{\int_0^{\infty} e^{-\frac{1}{2} y^2} d y\right\} \\
& =2 \int_0^{\infty} \int_0^{\infty} e^{-\frac{1}{2}\left(x^2+y^2\right)} d x d y .
\end{aligned}
$$
Now use polar coordinates to evaluate the double integral and thereby show that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$.
51. [15] Get High Airlines wants to estimate the fraction of smokers $p$ among their passenger population. The airline plans to use sampling with replacement to determine their estimate $\hat{p}$ of $p$. They set up the requirement that
$$
P[|\hat{p}-p| \geq 0.005] \leq 0.05 \text {. }
$$
(a) How large a sample should Get High take if nothing is assumed about the size of $p$ ?
Assume $\rho$ is near 0.5 .

Key Concepts

Gamma Function and Its Properties

The gamma function extends the factorial function to non-integer values and is defined via an improper integral. It possesses a recurrence relation, ?(t+1) = t?(t), which generalizes the notion of successive multiplication found in factorials. Additionally, through variable transformations and utilizing polar coordinates, the gamma function is linked with integrals that arise in probability theory and statistics, such as the integral representations of the normal distribution, exemplified by the evaluation of ?(1/2) in terms of ?.

Student's t Distribution Properties

The Student's t distribution arises in the context of estimating means from normally distributed populations when the sample size is small and the variance is unknown. A key property is its unimodality: the density function reaches a unique maximum at x = 0, reflecting the symmetric and bell-shaped nature of the distribution. This property is important in understanding inference based on the t-distribution, particularly in hypothesis testing and constructing confidence intervals.

Sample Size Determination for Proportion Estimates

Determining an appropriate sample size for estimating a population proportion involves ensuring that the estimator falls within a desired error bound with a specified confidence level. When little is known about the proportion in advance, a worst-case scenario (often assuming a proportion near 0.5) is used because it maximizes the variance. This concept is crucial for designing surveys and experiments where the objective is to achieve reliable and precise estimation with minimal sampling error.

Estimation of Proportions through Simulation

This concept involves using a large number of random samples to estimate the probability or proportion of a particular event — such as a system component being busy. By taking repeated independent measurements and computing the ratio of successes (or the event occurring) to total samples, one approximates the true probability. Understanding this is critical in simulation studies where the goal is to ensure that the estimated proportion is close to the actual performance metric.

Concentration Inequalities

Concentration inequalities, such as Chebyshev's inequality or Hoeffding's inequality, provide bounds on the probability that a sum of random variables deviates from its expected value by a given amount. They are integral in determining the number of samples required to achieve a certain margin of error with high confidence, which is essential when setting specifications like ? (tolerance level) and ? (error probability) in simulation and sampling experiments.

Transcript

00:01 For this exercise, we are given the following information.

00:05 So we have these hypotheses, and we're told that the true parameter p is p prime, and p prime is less than the null hypothesized proportion.

00:19 So therefore, the alternative hypothesis is actually true.

00:25 Now, for part a, we are asked to show the following, where z is the test statistic for a one -proportion test.

00:32 So the 1 proportion test statistic is given as follows.

00:55 Now if we calculate the expectation on z, everything inside the brackets is constant except for the sample proportion.

01:27 So the hypothesized proportion and n are both constant.

01:32 So the expected value can be rewritten like this.

01:45 This is because the expected value for a constant is that constant.

02:04 Now we also know from chapter 2 that the expected value for a constant.

02:09 For a sample proportion is the true value.

02:21 So this becomes, and this is what we are trying to show, at least with respect to the expected value.

02:53 So now for the variance, now in our test statistic, again, everything is a constant except for the sample proportion.

03:07 So that is the only varying parameter.

03:20 I'll rewrite this like this.

03:57 So i'm taking this constant here outside of the variance brackets.

04:05 So therefore i squared.

04:07 So that's why the square root symbol has disappeared.

04:14 And we also know from chapter 2 that the variance for a sample proportion is given by the following...

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