91Ó°ÊÓ

At State University, the average score of the entering class on the verbal portion of the SAT is 565 , with a standard deviation of 75 . Marian scored a 660 . How many of State's other 4250 freshmen did better? Assume that the scores are normally distributed.

The IQs of nine randomly selected people are recorded. Let \(\bar{Y}\) denote their average. Assuming the distribution from which the \(Y_{i}\) 's were drawn is normal with a mean of 100 and a standard deviation of 16 , what is the probability that \(\bar{Y}\) will exceed 103 ? What is the probability that any arbitrary \(Y_{i}\) will exceed 103 ? What is the probability that exactly three of the \(Y_{i}\) 's will exceed 103 ?

Because of her past convictions for mail fraud and forgery, Jody has a \(30 \%\) chance each year of having her tax returns audited. What is the probability that she will escape detection for at least three years? Assume that she exaggerates, distorts, misrepresents, lies, and cheats every year.

Show that the cdf for a geometric random variable is given by \(F_{X}(t)=P(X \leq t)=1-(1-p)^{[r]}\), where \([t]\) denotes the greatest integer in \(t, t \geq 0\).

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?

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 \(X_{.)}\)

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?