Chapter 3

Let \(Y\) be a continuous nonnegative random variable. Show that \(W=Y^{2}\) has pdf \(f_{W}(w)=\frac{1}{2 \sqrt{w}} f_{Y}(\sqrt{w})\). (Hint: First find \(F_{W}(w) .\) )

Let \(Y\) be a uniform random variable over the interval \([0,1]\). Find the pdf of \(W=Y^{2}\).

Let \(Y\) be a random variable with \(f_{Y}(y)=6 y(1-y)\), \(0 \leq y \leq 1\). Find the pdf of \(W=Y^{2}\).

Suppose the velocity of a gas molecule of mass \(m\) is a random variable with pdf \(f_{Y}(y)=a y^{2} e^{-b y^{2}}, y \geq 0\), where \(a\) and \(b\) are positive constants depending on the gas. Find the pdf of the kinetic energy, \(W=(m / 2) Y^{2}\), of such a molecule.

Let \(X\) and \(Y\) be two independent random variables. Given the marginal pdfs indicated below, find the cdf of \(Y / X\). (Hint: Consider two cases, \(0 \leq w \leq 1\) and \(1

Suppose that \(X\) and \(Y\) are two independent random variables, where \(f_{X}(x)=x e^{-x}, x \geq 0\), and \(f_{Y}(y)=\) \(e^{-y}, y \geq 0\). Find the pdf of \(Y / X\).

Suppose that \(r\) chips are drawn with replacement from an urn containing \(n\) chips, numbered 1 through \(n\). Let \(V\) denote the sum of the numbers drawn. Find \(E(V)\).

Suppose that \(f_{X, Y}(x, y)=\lambda^{2} e^{-\lambda(x+y)}, 0 \leq x, 0 \leq y\). Find \(E(X+Y)\).

Marksmanship competition at a certain level requires each contestant to take ten shots with each of two different handguns. Final scores are computed by taking a weighted average of four times the number of bull's-eyes made with the first gun plus six times the number gotten with the second. If Cathie has a \(30 \%\) chance of hitting the bull's-eye with each shot from the first gun and a \(40 \%\) chance with each shot from the second gun, what is her expected score?

An urn contains \(r\) red balls and \(w\) white balls. A sample of \(n\) balls is drawn in order and without replacement. Let \(X_{i}\) be 1 if the \(i\) th draw is red and 0 otherwise, \(i=1,2, \ldots, n .\) (a) Show that \(E\left(X_{i}\right)=E\left(X_{1}\right), i=2,3, \ldots, n\). (b) Use the corollary to Theorem \(3.9 .2\) to show that the expected number of red balls is \(n r /(r+w)\).

Suppose two fair dice are tossed. Find the expected value of the product of the faces showing.

Suppose \(X\) represents a point picked at random from the interval \([0,1]\) on the \(x\)-axis, and \(Y\) is a point picked at random from the interval \([0,1]\) on the \(y\)-axis. Assume that \(X\) and \(Y\) are independent. What is the expected value of the area of the triangle formed by the points \((X, 0),(0, Y)\), and \((0,0)\) ?

Suppose \(Y_{1}, Y_{2}, \ldots, Y_{n}\) is a random sample from the uniform pdf over \([0,1]\). The geometric mean of the numbers is the random variable \(\sqrt[n]{Y_{1} Y_{2} \cdots \cdot Y_{n}}\). Compare the expected value of the geometric mean to that of the arithmetic mean \(\bar{Y}\).

Suppose that two dice are thrown. Let \(X\) be the number showing on the first die and let \(Y\) be the larger of the two numbers showing. Find \(\operatorname{Cov}(X, Y)\).

Show that $$ \operatorname{Cov}(a X+b, c Y+d)=a c \operatorname{Cov}(X, Y) $$ for any constants \(a, b, c\), and \(d\).

Let \(U\) be a random variable uniformly distributed over \([0,2 \pi]\). Define \(X=\cos U\) and \(Y=\sin U\). Show that \(X\) and \(Y\) are dependent but that \(\operatorname{Cov}(X, Y)=0\).

Let \(X\) and \(Y\) be random variables with $$ f_{X, Y}(x, y)=\left\\{\begin{array}{ll} 1, & -y

Suppose that \(f_{X, Y}(x, y)=\lambda^{2} e^{-2(x+y)}, 0 \leq x, 0 \leq y\). Find \(\operatorname{Var}(X+Y)\). (Hint: See Questions 3.6.11 and 3.9.2.)

Suppose that \(f_{X, Y}(x, y)=\frac{2}{2}\left(x^{2}+y^{2}\right), 0 \leq x \leq 1\), \(0 \leq y \leq 1\). Find \(\operatorname{Var}(X+Y)\).

Let \(X\) be a binomial random variable based on \(n\) trials and a success probability of \(p_{X}\); let \(Y\) be an independent binomial random variable based on \(m\) trials and a success probability of \(p_{Y}\). Find \(E(W)\) and \(\operatorname{Var}(W)\), where \(W=4 X+6 Y\)

A Poisson random variable has pdf \(p_{X}(k)=e^{-\lambda} \frac{\lambda^{k}}{k !}\), \(k=0,1,2, \ldots\) and \(\lambda>0\) (see Section 4.2). Also, \(E(X)=\lambda\). Suppose the Poisson random variable \(U\) is the number of calls for technical assistance received by a computer company during the firm's nine normal workday hours, with the average number of calls per hour equal 70 . Also suppose each call costs the company \(\$ 50\). Let \(V\) be a Poisson random variable representing the number of calls for technical assistance received during a day's remaining fifteen hours. Assume the average number of calls per hour is four for that time period and that each such call costs the company \(\$ 60\). Find the expected cost and the variance of the cost associated with the calls received during a twenty-four-hour day.

A mason is contracted to build a patio retaining wall. Plans call for the base of the wall to be a row of fifty 10 -inch bricks, each separated by \(\frac{1}{2}\)-inch-thick mortar. Suppose that the bricks used are randomly chosen from a population of bricks whose mean length is 10 inches and whose standard deviation is \(\frac{1}{32}\) inch. Also, suppose that the mason, on the average, will make the mortar \(\frac{1}{2}\) inch thick, but that the actual dimension will vary from brick to brick, the standard deviation of the thicknesses being \(\frac{1}{16}\) inch. What is the standard deviation of \(L\), the length of the first row of the wall? What assumption are you making?

A gambler plays \(n\) hands of poker. If he wins the \(k\) th hand, he collects \(k\) dollars; if he loses the \(k\) th hand, he collects nothing. Let \(T\) denote his total winnings in \(n\) hands. Assuming that his chances of winning each hand are constant and independent of his success or failure at any other hand, find \(E(T)\) and \(\operatorname{Var}(T)\).

. Suppose the length of time, in minutes, that you have to wait at a bank teller's window is uniformly distributed over the interval \((0,10)\). If you go to the bank four times during the next month, what is the probability that your second longest wait will be less than five minutes?

. A random sample of size \(n=6\) is taken from the pdf \(f_{Y}(y)=3 y^{2}, 0 \leq y \leq 1\). Find \(P\left(Y_{5}^{\prime}>0.75\right)\).

What is the probability that the larger of two random observations drawn from any continuous pdf will exceed the sixtieth percentile?

A random sample of size 5 is drawn from the pdf \(f_{Y}(y)=2 y, 0 \leq y \leq 1\). Calculate \(P\left(Y_{1}^{\prime}<0.6

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be a random sample from the exponential pdf \(f_{y}(y)=e^{-y}, y \geq 0\). What is the smallest \(n\) for which \(P\left(Y_{\min }<0.2\right)>0.9\) ?

Calculate \(P\left(0.6

Suppose that \(n\) observations are chosen at ran dom from a continuous pdf \(f_{Y}(y)\). What is the probabil ity that the last observation recorded will be the smalles number in the entire sample?

Consider a system containing \(n\) components, where the lifetimes of the components are independent random variables and each has pdf \(f_{Y}(y)=\lambda e^{-\lambda y}\), \(y>0\). Show that the average time elapsing before the first component failure occurs is \(1 / n \lambda\).

Suppose three points are picked randomly from the unit interval. What is the probability that the three are within a half unit of one another?

Suppose \(X\) and \(Y\) have the joint pdf \(p_{X, Y}(x, y)=\) \(\frac{x+y+x y}{21}\) for the points \((1,1),(1,2),(2,1),(2,2)\), where \(X\) denotes a "message" sent (either \(x=1\) or \(x=2\) ) and \(Y\) denotes a "message" received. Find the probability that the message sent was the message received, that is, find \(p_{Y \mid x}(x)\).

Suppose a die is rolled six times. Let \(X\) be the total number of 4 's that occur and let \(Y\) be the number of 4 's in the first two tosses. Find \(p_{Y \mid x}(y)\).

An urn contains eight red chips, six white chips, and four blue chips. A sample of size 3 is drawn without replacement. Let \(X\) denote the number of red chips in the sample and \(Y\), the number of white chips. Find an expression for \(p_{Y \mid x}(y)\).

Five cards are dealt from a standard poker deck. Let \(X\) be the number of aces received, and \(Y\) the number of kings. Compute \(P(X=2 \mid Y=2)\).

Given that two discrete random variables \(X\) and \(Y\) follow the joint pdf \(p_{X, Y}(x, y)=k(x+y)\), for \(x=1,2,3\) and \(y=1,2,3\), (a) Find \(k\). (b) Evaluate \(p_{Y \mid x}(1)\) for all values of \(x\) for which \(p_{x}(x)>0\).

Let \(X\) denote the number on a chip drawn at random from an urn containing three chips, numbered 1,2, and 3 . Let \(Y\) be the number of heads that occur when a fair coin is tossed \(X\) times.

Suppose \(X, Y\), and \(Z\) have a trivariate distribution described by the joint pdf $$ p_{X, Y, Z}(x, y, z)=\frac{x y+x z+y z}{54} $$ where \(x, y\), and \(z\) can be 1 or 2 . Tabulate the joint conditional pdf of \(X\) and \(Y\) given each of the two values of \(z\).

Let \(X\) be a nonnegative random variable. We say that \(X\) is memoryless if $$ P(X>s+t \mid X>t)=P(X>s) \text { for all } s, t \geq 0 $$ Show that a random variable with pdf \(f_{X}(x)=(1 / \lambda) e^{-x / \lambda}\), \(x>0\), is memoryless.

Given the joint pdf $$ f_{X, Y}(x, y)=2 e^{-(x+y)}, \quad 0 \leq x \leq y, \quad y \geq 0 $$ find (a) \(P(Y<1 \mid X<1)\). (b) \(P(Y<1 \mid X=1)\). (c) \(f_{Y \mid x}(y)\). (d) \(E(Y \mid x)\).

Find the conditional pdf of \(Y\) given \(x\) if $$ f_{X, Y}(x, y)=x+y $$ for \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\).

\( If $$ f_{X, Y}(x, y)=2, \quad x \geq 0, \quad y \geq 0, \quad x+y \leq 1 $$ show that the conditional pdf of \)Y\( given \)x$ is uniform.

Suppose that $$ f_{Y \mid x}(y)=\frac{2 y+4 x}{1+4 x} \quad \text { and } \quad f_{X}(x)=\frac{1}{3} \cdot(1+4 x) $$ for \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\). Find the marginal pdf for \(Y\).

Suppose that \(X\) and \(Y\) are distributed according to the joint pdf $$ f_{X, Y}(x, y)=\frac{2}{5} \cdot(2 x+3 y), \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1 $$ Find (a) \(f_{X}(x)\). (b) \(f_{Y \mid x}(y)\). (c) \(P\left(\frac{1}{4} \leq Y \leq \frac{3}{4} \mid X=\frac{1}{2}\right)\). (d) \(E(Y \mid x)\).

If \(X\) and \(Y\) have the joint pdf $$ f_{X, Y}(x, y)=2, \quad 0 \leq x

Find \(P\left(X<1 \mid Y=1 \frac{1}{2}\right)\) if \(X\) and \(Y\) have the joint pdf $$ f_{X, Y}(x, y)=x y / 2, \quad 0 \leq x

Suppose that \(X_{1}, X_{2}, X_{3}, X_{4}\), and \(X_{5}\) have the joint pdf $$ f x_{1}, X_{2}, X_{3}, X_{4}, X_{5}\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=32 x_{1} x_{2} x_{3} x_{4} x_{5} $$ for \(0 \leq x_{i} \leq 1, i=1,2, \ldots, 5\). Find the joint conditional pdf of \(X_{1}, \bar{X}_{2}\), and \(X_{3}\) given that \(X_{4}=x_{4}\) and \(X_{5}=x_{5}\).

Suppose the random variables \(X\) and \(Y\) are jointly distributed according to the pdf $$ f_{X, Y}(x, y)=\frac{6}{7}\left(x^{2}+\frac{x y}{2}\right), \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2 $$ Find (a) \(f_{X}(x)\). (b) \(P(X>2 Y)\). (c) \(P\left(Y>1 \mid X>\frac{1}{2}\right)\).

For continuous random variables \(X\) and \(Y\), prove that \(E(Y)=E_{X}[E(Y \mid x)]\).