Chapter 4

Sometimes the geometric random variable is defined to be the number of trials, \(X\), preceding the first success. Write down the corresponding pdf and derive the moment-generating function for \(X\) two ways(1) by evaluating \(E\left(e^{2 X}\right)\) directly and (2) by using Theorem \(3.12 .3\).

Suppose that the random variables \(X_{1}\) and \(X_{2}\) have \(\mathrm{mgfs} M_{X_{1}}(t)=\frac{\frac{1}{3 e^{2}}}{1-\left(1-\frac{1}{2}\right)}\) and \(M_{X_{2}}(t)=\frac{\frac{1}{2} e}{1-\left(1-\frac{1}{4} t\right)}\), respectively. Let \(X=X_{1}+X_{2}\). Does \(X\) have a geometric distribution? Assume that \(X_{1}\) and \(X_{2}\) are independent.

A door-to-door encyclopedia salesperson is required to document five in-home visits each day. Suppose that she has a \(30 \%\) chance of being invited into any given home, with each address representing an independent trial. What is the probability that she requires fewer than eight houses to achieve her fifth success?

An underground military installation is fortified to the extent that it can withstand up to three direct hits from air-to-surface missiles and still function. Suppose an enemy aircraft is armed with missiles, each having a \(30 \%\) chance of scoring a direct hit. What is the probability that the installation will be destroyed with the seventh missile fired?

Darryl's statistics homework last night was to flip a fair coin and record the toss, \(X\), when heads appeared for the second time. The experiment was to be repeated a total of one hundred times. The following are the one hundred values for \(X\) that Darryl turned in this morning. Do you think that he actually did the assignment? Explain. $$ \begin{array}{rrrrrrrrrr} 3 & 7 & 3 & 2 & 9 & 3 & 4 & 3 & 3 & 2 \\ 7 & 3 & 8 & 4 & 3 & 3 & 3 & 4 & 3 & 3 \\ 4 & 3 & 2 & 2 & 4 & 5 & 2 & 2 & 2 & 4 \\ 2 & 5 & 6 & 4 & 2 & 6 & 2 & 8 & 3 & 2 \\ 8 & 2 & 3 & 2 & 4 & 3 & 2 & 6 & 3 & 3 \\ 3 & 2 & 5 & 3 & 6 & 4 & 5 & 6 & 5 & 6 \\ 3 & 5 & 2 & 7 & 2 & 10 & 4 & 3 & 2 & 2 \\ 4 & 2 & 4 & 5 & 5 & 5 & 6 & 2 & 4 & 3 \\ 3 & 4 & 4 & 6 & 3 & 4 & 2 & 5 & 5 & 2 \\ 5 & 7 & 5 & 3 & 2 & 7 & 4 & 4 & 4 & 3 \end{array} $$

When a machine is improperly adjusted, it has probability \(0.15\) of producing a defective item. Each day, the machine is run until three defective items are produced. When this occurs, it is stopped and checked for adjustment. What is the probability that an improperly adjusted machine will produce five or more items before being stopped? What is the average number of items an improperly adjusted machine will produce before being stopped?

Let the random variable \(X\) denote the number of trials in excess of \(r\) that are required to achieve the \(r\) th success in a series of independent trials, where \(p\) is the probability of success at any given trial. Show that \(p X(k)=\left(\begin{array}{c}k+r-1 \\ k\end{array}\right) p^{\prime}(1-p)^{k}, \quad k=0,1,2, \ldots\) (Note: This particular formula for \(p_{X}(k)\) is often used in place of Equation \(4.5 .1\) as the definition of the pdf for a negative binomial random variable.)

Let \(X_{1}, X_{2}\), and \(X_{3}\) be three independent negative binomial random variables with pdfs $$ p_{X_{i}}(k)=\left(\begin{array}{c} k-1 \\ 2 \end{array}\right)\left(\frac{4}{5}\right)^{3}\left(\frac{1}{5}\right)^{k-3}, \quad k=3,4,5, \ldots $$ for \(i=1,2,3 .\) Define \(X=X_{1}+X_{2}+X_{3} .\) Find \(P(10 \leq\) \(X \leq 12)\). (Hint: Use the moment-generating functions of \(X_{1}, X_{2}\), and \(X_{3}\) to deduce the pdf of \(X_{.)}\)

Suppose that \(X_{1}, X_{2}, \ldots, X_{k}\) are independent negative binomial random variables with parameters \(r_{1}\) and \(p, r_{2}\) and \(p, \ldots\), and \(r_{k}\) and \(p\), respectively. Let \(X=X_{1}+\) \(X_{2}+\cdots+X_{k} .\) Find \(M_{X}(t), p_{X}(t), E(X)\), and \(\operatorname{Var}(X)\)

An Arctic weather station has three electronic wind gauges. Only one is used at any given time. The lifetime of each gauge is exponentially distributed with a mean of one thousand hours. What is the pdf of \(Y\), the random variable measuring the time until the last gauge wears out?

Demonstrate that \(\lambda\) plays the role of a scale parameter by showing that if \(Y\) is gamma with parameters \(r\) and \(\lambda\), then \(\lambda Y\) is gamma with parameters \(r\) and \(1 .\)

Show that a gamma pdf has the unique mode \(\frac{r-1}{\lambda}\); that is, show that the function \(f_{Y}(y)=\frac{\alpha^{\prime}}{\Gamma(r)} y^{\prime-1} e^{-\lambda y}\) takes its maximum value at \(y_{\text {mode }}=\frac{r-1}{\lambda}\) and at no other point.

Prove that \(\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\). (Hint: Consider \(E\left(Z^{2}\right)\), where \(Z\) is a standard normal random variable.)

Show that \(\Gamma\left(\frac{7}{2}\right)=\frac{15}{8} \sqrt{\pi}\).

If the random variable \(Y\) has the gamma pdf with integer parameter \(r\) and arbitrary \(\lambda>0\), show that $$ E\left(Y^{m}\right)=\frac{(m+r-1) !}{(r-1) ! \lambda^{m}} $$ (Hint: Use the fact that \(\int_{0}^{\infty} y^{r-1} e^{-y} d y=(r-1) !\) when \(r\) is a positive integer.)