Chapter 3

Recall the game of Keno described in Question 3.2.26. The following are all the payoffs on a \(\$ 1\) wager where the player has bet on ten numbers. Calculate \(E(X)\), where the random variable \(X\) denotes the amount of money won. \begin{tabular}{crc} \hline Number of Correct Guesses & Payoff & Probability \\ \hline\(<5\) & \(-81\) & \(.935\) \\ 5 & 2 & \(.0514\) \\ 6 & 18 & \(.0115\) \\ 7 & 180 & \(.0016\) \\ 8 & 1,300 & \(1.35 \times 10^{-4}\) \\ 9 & 2,600 & \(6.12 \times 10^{-6}\) \\ 10 & 10,000 & \(1.12 \times 10^{-7}\) \\ \hline \end{tabular}

In the game of redball, two drawings are made without replacement from a bowl that has four white pingpong balls and two red ping-pong balls. The amount won is determined by how many of the red balls are selected. For a \(\$ 5\) bet, a player can opt to be paid under either Rule \(A\) or Rule \(B\), as shown. If you were playing the game, which would you choose? Why? \begin{tabular}{lcccc} \hline \multicolumn{2}{c}{\(A\)} & & \multicolumn{2}{c}{\(B\)} \\ \cline { 4 - 5 } No. of Red Balls Drawn & Payoff & & No. of Red Balls Drawn & Payoff \\ \hline 0 & 0 & & 0 & 0 \\ 1 & \(\$ 2\) & & 1 & \(\$ 1\) \\ 2 & \(\$ 10\) & & 2 & \(\$ 20\) \\ \hline \end{tabular}

Suppose a life insurance company sells a \(\$ 50,000\), five-year term policy to a twenty-five-year-old woman. At the beginning of each year the woman is alive, the company collects a premium of \(\$ P\). The probability that the woman dies and the company pays the \(\$ 50,000\) is given in the table below. So, for example, in Year 3 , the company loses \(\$ 50,000-\$ P\) with probability \(0.00054\) and gains \(\$ P\) with probability \(1-0.00054=0.99946\). If the company expects to make \(\$ 1000\) on this policy, what should \(P\) be? \begin{tabular}{cc} \hline Year & Probability of Payoff \\ \hline 1 & \(0.00051\) \\ 2 & \(0.00052\) \\ 3 & \(0.00054\) \\ 4 & \(0.00056\) \\ 5 & \(0.00059\) \\ \hline \end{tabular}

A manufacturer has one hundred memory chips in stock, \(4 \%\) of which are likely to be defective (based on past experience). A random sample of twenty chips is selected and shipped to a factory that assembles laptops. Let \(X\) denote the number of computers that receive faulty memory chips. Find \(E(X)\).

Records show that 642 new students have just entered a certain Florida school district. Of those 642 , a total of 125 are not adequately vaccinated. The district's physician has scheduled a day for students to receive whatever shots they might need. On any given day, though, \(12 \%\) of the district's students are likely to be absent. How many new students, then, can be expected to remain inadequately vaccinated?

Calculate \(E(Y)\) for the following pdfs: (a) \(f_{Y}(y)=3(1-y)^{2}, 0 \leq y \leq 1\) (b) \(f_{Y}(y)=4 y e^{-2 y}, y \geq 0\) (c) \(f_{Y}(y)= \begin{cases}\frac{3}{4}, & 0 \leq y \leq 1 \\ \frac{1}{4}, & 2 \leq y \leq 3 \\ 0, & \text { elsewhere }\end{cases}\) (d) \(f_{Y}(y)=\sin y, \quad 0 \leq y \leq \frac{\pi}{2}\)

Recall Question \(3.4 .4\), where the length of time \(Y\) (in years) that a malaria patient spends in remission has pdf \(f_{Y}(y)=\frac{1}{9} y^{2}, 0 \leq y \leq 3\). What is the average length of time that such a patient spends in remission?

Let the random variable \(Y\) have the uniform distribution over \([a, b]\); that is, \(f_{Y}(y)=\frac{1}{b-a}\) for \(a \leq y \leq b\). Find \(E(Y)\) using Definition 3.5.1. Also, deduce the value of \(E(Y)\), knowing that the expected value is the center of gravity of \(f_{Y}(y)\).

Show that the expected value associated with the exponential distribution, \(f_{Y}(y)=\lambda e^{-\lambda y}, y>0\), is \(1 / \lambda\), where \(\lambda\) is a positive constant.

Show that $$ f_{Y}(y)=\frac{1}{y^{2}}, \quad y \geq 1 $$ is a valid pdf but that \(Y\) does not have a finite expected value.

Based on recent experience, ten-year-old passenger cars going through a motor vehicle inspection station have an \(80 \%\) chance of passing the emissions test. Suppose that two hundred such cars will be checked out next week. Write two formulas that show the number of cars that are expected to pass.

Suppose that fifteen observations are chosen at random from the pdf \(f_{Y}(y)=3 y^{2}, 0 \leq y \leq 1\). Let \(X\) denote the number that lie in the interval \(\left(\frac{1}{2}, 1\right)\). Find \(E(X)\).

A city has 74,806 registered automobiles. Each is required to display a bumper decal showing that the owner paid an annual wheel tax of \(\$ 50\). By law, new decals need to be purchased during the month of the owner's birthday. How much wheel tax revenue can the city expect to receive in November?

An urn contains four chips numbered 1 through 4 . Two are drawn without replacement. Let the random variable \(X\) denote the larger of the two. Find \(E(X)\).

A fair coin is tossed three times. Let the random variable \(X\) denote the total number of heads that appear times the number of heads that appear on the first and third tosses. Find \(E(X)\).

How much would you have to ante to make the St. Petersburg game "fair" (recall Example 3.5.5) if the most you could win was \(\$ 1000 ?\) That is, the payoffs are \(\$ 2^{k}\) for \(1 \leq k \leq 9\), and \(\$ 1000\) for \(k \geq 10\).

For the St. Petersburg problem (Example 3.5.5), find the expected payoff if (a) the amounts won are \(c^{k}\) instead of \(2^{k}\), where \(0<\) \(c<2\). (b) the amounts won are \(\log 2^{k}\). [This was a modification suggested by D. Bernoulli (a nephew of James Bernoulli) to take into account the decreasing marginal utility of money \(-\) the more you have, the less useful a bit more is.]

A fair die is rolled three times. Let \(X\) denote the number of different faces showing, \(X=1,2,3\). Find \(E(X)\).

Two distinct integers are chosen at random from the first five positive integers. Compute the expected value of the absolute value of the difference of the two numbers.

Suppose that two evenly matched teams are playing in the World Series. On the average, how many games will be played? (The winner is the first team to get four victories.) Assume that each game is an independent event.

An urn contains one white chip and one black chip. A chip is drawn at random. If it is white, the "game" is over; if it is black, that chip and another black one are put into the urn. Then another chip is drawn at random from the "new" urn and the same rules for ending or continuing the game are followed (i.e., if the chip is white, the game is over; if the chip is black, it is placed back in the urn, together with another chip of the same color). The drawings continue until a white chip is selected. Show that the expected number of drawings necessary to get a white chip is not finite.

. A random sample of size \(n\) is drawn without replacement from an urn containing \(r\) red chips and \(w\) white chips. Define the random variable \(X\) to be the number of red chips in the sample. Use the summation technique described in Theorem 3.5.1 to prove that \(E(X)=\) \(r n /(r+w)\)

Given that \(X\) is a nonnegative, integer-valued random variable, show that $$ E(X)=\sum_{k=1}^{\infty} P(X \geq k) $$

Find the median for each of the following pdfs: (a) \(f_{Y}(y)=(\theta+1) y^{\theta}, 0 \leq y \leq 1\), where \(\theta>0\) (b) \(f_{Y}(y)=y+\frac{1}{2}, 0 \leq y \leq 1\)

Suppose \(X\) is a binomial random variable with \(n=10\) and \(p=\frac{2}{5}\). What is the expected value of \(3 X-4\) ?

A typical day's production of a certain electronic component is twelve. The probability that one of these components needs rework is \(0.11\). Each component needing rework costs \(\$ 100\). What is the average daily cost for defective components?

Let \(Y\) have probability density function $$ f_{Y}(y)=2(1-y), 0 \leq y \leq 1 $$ Suppose that \(W=Y^{2}\), in which case $$ f_{W}(w)=\frac{1}{\sqrt{w}}-1,0 \leq w \leq 1 $$ Find \(E(W)\) in two different ways.

. A tool and die company makes castings for steel stress-monitoring gauges. Their annual profit, \(Q\), in hundreds of thousands of dollars, can be expressed as a function of product demand, \(y\) : $$ Q(y)=2\left(1-e^{-2 y}\right) $$ Suppose that the demand (in thousands) for their castings follows an exponential pdf, \(f_{Y}(y)=6 e^{-6 y}, y>0\). Find the company's expected profit.

A box is to be constructed so that its height is five inches and its base is \(Y\) inches by \(Y\) inches, where \(Y\) is a random variable described by the pdf, \(f_{Y}(y)=6 y(1-y)\), \(0

Grades on the last Economics 301 exam were not very good. Graphed, their distribution had a shape similar to the pdf $$ f_{Y}(y)=\frac{1}{5000}(100-y), \quad 0 \leq y \leq 100 $$ As a way of "curving" the results, the professor announces that he will replace each person's grade, \(Y\), with a new grade, \(g(Y)\), where \(g(Y)=10 \sqrt{Y}\). Will the professor's strategy be successful in raising the class average above \(60 ?\)

If \(Y\) has probability density function $$ f_{Y}(y)=2 y, 0 \leq y \leq 1 $$ then \(E(Y)=\frac{2}{3}\). Define the random variable \(W\) to be the squared deviation of \(Y\) from its mean, that is, \(W=\) \(\left(Y-\frac{2}{3}\right)^{2}\). Find \(E(W)\).

Use Theorem \(3.6 .1\) to find the variance of the random variable \(Y\), where $$ f_{Y}(y)=3(1-y)^{2}, \quad 0

An urn contains \(n\) chips numbered 1 through \(n\). Assume that the probability of choosing chip \(i\) is equal to \(k i, i=1,2, \ldots, n\). If one chip is drawn, calculate \(E\left(\frac{1}{X}\right)\), where the random variable \(X\) denotes the number showing on the chip selected. [Hint: Recall that the sum of the first \(n\) integers is \(n(n+1) / 2\).]

Find the variance of \(Y\) if $$ f_{Y}(y)= \begin{cases}\frac{3}{4}, & 0 \leq y \leq 1 \\ \frac{1}{4}, & 2 \leq y \leq 3 \\ 0, & \text { elsewhere }\end{cases} $$

Ten equally qualified applicants, six men and four women, apply for three lab technician positions. Unable to justify choosing any of the applicants over all the others, the personnel director decides to select the three at random. Let \(X\) denote the number of men hired. Compute the standard deviation of \(X\).

If $$ f_{Y}(y)=\frac{2 y}{k^{2}}, \quad 0 \leq y \leq k $$ for what value of \(k\) does \(\operatorname{Var}(Y)=2 ?\)

Frankie and Johnny play the following game. Frankie selects a number at random from the interval \([a, b]\). Johnny, not knowing Frankie's number, is to pick a second number from that same interval and pay Frankie an amount, \(W\), equal to the squared difference between

Suppose that \(Y\) is an exponential random variable, so \(f_{Y}(y)=\lambda e^{-\lambda y}, y \geq 0\). Show that the variance of \(Y\) is \(1 / \lambda^{2}\).

Suppose that \(Y\) is an exponential random variable with \(\lambda=2\) (recall Question 3.6.11). Find \(P[Y>\) \(E(Y)+2 \sqrt{\operatorname{Var}(Y)}] .\)

Let \(X\) be a random variable with finite mean \(\mu\). Define for every real number \(a, g(a)=E\left[(X-a)^{2}\right]\). Show that $$ g(a)=E\left[(X-\mu)^{2}\right]+(\mu-a)^{2} . $$ What is another name for \(\min g(a)\) ?

Suppose the charge for repairing an automobile averages \(\$ 200\) with a standard deviation of \(\$ 16\). If a \(10 \%\) \(\operatorname{tax}\) is added to the charge and then a \(\$ 15\) flat fee for environmental impact, what is the standard deviation of the charge to the car owner?

If \(E(W)=\mu\) and \(\operatorname{Var}(W)=\sigma^{2}\), show that \(E\left(\frac{W-\mu}{\sigma}\right)=0\) and \(\operatorname{Var}\left(\frac{W-\mu}{\sigma}\right)=1\)

. Suppose \(U\) is a uniform random variable over \([0,1]\). (a) Show that \(Y=(b-a) U+a\) is uniform over \([a, b]\). (b) Use part (a) and Theorem \(3.6 .2\) to find the variance of \(Y\).

Recovering small quantities of calcium in the presence of magnesium can be a difficult problem for an analytical chemist. Suppose the amount of calcium \(Y\) to be recovered is uniformly distributed between 4 and \(7 \mathrm{mg}\). The amount of calcium recovered by one method is the random variable $$ W_{1}=0.2281+(0.9948) Y+E_{1} $$ where the error term \(E_{1}\) has mean 0 and variance \(0.0427\) and is independent of \(Y\). A second procedure has random variable $$ W_{2}=-0.0748+(1.0024) Y+E_{2} $$ where the error term \(E_{2}\) has mean 0 and variance \(0.0159\) and is independent of \(Y\). The better technique should have a mean as close as possible to the mean of \(Y(=5.5)\), and a variance as small as possible. Compare the two methods on the basis of mean and variance.

Let \(Y\) be a uniform random variable defined over the interval \((0,2)\). Find an expression for the \(r\) th moment of \(Y\) about the origin. Also, use the binomial expansion as described in the Comment to find \(E\left[(Y-\mu)^{6}\right]\).

. Find the coefficient of skewness for an exponential random variable having the pdf $$ f_{Y}(y)=e^{-y}, \quad y>0 $$

Calculate the coefficient of kurtosis for a uniform random variable defined over the unit interval, \(f_{Y}(y)=1\), for \(0 \leq y \leq 1\).

Suppose that \(W\) is a random variable for which \(E\left[(W-\mu)^{3}\right]=10\) and \(E\left(W^{3}\right)=4\). Is it possible that \(\mu=2\) ?

Suppose that the random variable \(Y\) is described by the pdf $$ f_{Y}(y)=c \cdot y^{-6}, \quad y>1 $$ (a) Find \(c\). (b) What is the highest moment of \(Y\) that exists?

If \(p_{X, Y}(x, y)=c x y\) at the points \((1,1),(2,1),(2,2)\), and \((3,1)\), and equals 0 elsewhere, find \(c\).