Chapter 3

Suppose that random variables \(X\) and \(Y\) vary in accordance with the joint pdf, \(f_{X, Y}(x, y)=c(x+y), 0

Find \(c\) if \(f_{X, Y}(x, y)=c x y\) for \(X\) and \(Y\) defined over the triangle whose vertices are the points \((0,0),(0,1)\), and \((1,1)\).

An urn contains four red chips, three white chips, and two blue chips. A random sample of size 3 is drawn without replacement. Let \(X\) denote the number of white chips in the sample and \(Y\) the number of blue chips. Write a formula for the joint pdf of \(X\) and \(Y\).

Four cards are drawn from a standard poker deck. Let \(X\) be the number of kings drawn and \(Y\) the number of queens. Find \(p_{X, Y}(x, y)\).

Consider the experiment of tossing a fair coin three times. Let \(X\) denote the number of heads on the last flip, and let \(Y\) denote the total number of heads on the three flips. Find \(p_{X, Y}(x, y)\).

Suppose that two fair dice are tossed one time. Let \(X\) denote the number of 2 's that appear, and \(Y\) the number of 3 's. Write the matrix giving the joint probability density function for \(X\) and \(Y\). Suppose a third random variable, \(Z\), is defined, where \(Z=X+Y\). Use \(p_{X, Y}(x, y)\) to find \(p_{Z}(z)\).

Let \(X\) be the time in days between a car accident and reporting a claim to the insurance company. Let \(Y\) be the time in days between the report and payment of the claim. Suppose that \(f_{X, Y}(x, y)=c, 0 \leq x \leq 7,0 \leq y \leq 7\), and zero otherwise. (a) Find \(c\). (b) Find \(P(0 \leq X \leq 2,0 \leq Y \leq 4)\).

Let \(X\) and \(Y\) have the joint pdf $$ f_{X, Y}(x, y)=2 e^{-(x+y)}, \quad 0

A point is chosen at random from the interior of a circle whose equation is \(x^{2}+y^{2} \leq 4\). Let the random variables \(X\) and \(Y\) denote the \(x\) - and \(y\)-coordinates of the sampled point. Find \(f_{X, Y}(x, y)\).

Find \(P(X<2 Y)\) if \(f_{X, Y}(x, y)=x+y\) for \(X\) and \(Y\) each defined over the unit interval.

Suppose that five independent observations are drawn from the continuous pdf \(f_{T}(t)=2 t, 0 \leq t \leq 1\). Let \(X\) denote the number of \(t\) 's that fall in the interval \(0 \leq t<\frac{1}{3}\) and let \(Y\) denote the number of \(t\) 's that fall in the interval \(\frac{1}{3} \leq t<\frac{2}{3}\). Find \(p_{X, Y}(1,2)\).

A point is chosen at random from the interior of a right triangle with base \(b\) and height \(h\). What is the probability that the \(y\) value is between 0 and \(h / 2 ?\)

The campus recruiter for an international conglomerate classifies the large number of students she interviews into three categories - the lower quarter, the middle half, and the upper quarter. If she meets six students on a given morning, what is the probability that they will be evenly divided among the three categories? What is the marginal probability that exactly two will belong to the middle half?

For each of the following joint pdfs, find \(f_{X}(x)\) and \(f_{Y}(y)\). (a) \(f_{X, Y}(x, y)=\frac{1}{2}, 0 \leq x \leq 2,0 \leq y \leq 1\) (b) \(f_{X, Y}(x, y)=\frac{3}{2} y^{2}, 0 \leq x \leq 2,0 \leq y \leq 1\) (c) \(f_{X, Y}(x, y)=\frac{2}{3}(x+2 y), 0 \leq x \leq 1,0 \leq y \leq 1\) (d) \(f_{X, Y}(x, y)=c(x+y), 0 \leq x \leq 1,0 \leq y \leq 1\) (e) \(f_{X, Y}(x, y)=4 x y, 0 \leq x \leq 1,0 \leq y \leq 1\) (f) \(f_{X, Y}(x, y)=x y e^{-(x+y)}, 0 \leq x, 0 \leq y\) (g) \(f_{X, Y}(x, y)=y e^{-x y-y}, 0 \leq x, 0 \leq y\)

For each of the following joint pdfs, find \(f_{X}(x)\) and \(f_{Y}(y)\). (a) \(f_{X, Y}(x, y)=\frac{1}{2}, 0 \leq x \leq y \leq 2\) (b) \(f_{X, Y}(x, y)=\frac{1}{x}, 0 \leq y \leq x \leq 1\) (c) \(f_{X, Y}(x, y)=6 x, 0 \leq x \leq 1,0 \leq y \leq 1-x\)

Suppose that \(f_{X, Y}(x, y)=6(1-x-y)\) for \(x\) and \(y\) defined over the unit square, subject to the restriction that \(0 \leq x+y \leq 1\). Find the marginal pdf for \(X\).

. Suppose that \(X\) and \(Y\) are discrete random variables with $$ \begin{aligned} p_{X, Y}(x, y)=& \frac{4 !}{x ! y !(4-x-y) !}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{3}\right)^{y}\left(\frac{1}{6}\right)^{4-x-y}, \\\ & 0 \leq x+y \leq 4 \end{aligned} $$ Find \(p_{X}(x)\) and \(p_{Y}(x)\).

Consider the experiment of simultaneously tossing a fair coin and rolling a fair die. Let \(X\) denote the number of heads showing on the coin and \(Y\) the number of spots showing on the die. (a) List the outcomes in \(S\). (b) Find \(F_{X, Y}(1,2)\).

An urn contains twelve chips-four red, three black, and five white. A sample of size 4 is to be drawn without replacement. Let \(X\) denote the number of white chips in the sample, \(Y\) the number of red. Find \(F_{X, Y}(1,2)\).

For each of the following joint pdfs, find \(F_{X, Y}(x, y)\). (a) \(f_{X, Y}(x, y)=\frac{3}{2} y^{2}, 0 \leq x \leq 2,0 \leq y \leq 1\) (b) \(f_{X, Y}(x, y)=\frac{2}{3}(x+2 y), 0 \leq x \leq 1,0 \leq y \leq 1\) (c) \(f_{X, Y}(x, y)=4 x y, 0 \leq x \leq 1,0 \leq y \leq 1\)

For each of the following joint pdfs, find \(F_{X, Y}(x, y)\). (a) \(f_{X, Y}(x, y)=\frac{1}{2}, 0 \leq x \leq y \leq 2\) (b) \(f_{X, Y}(x, y)=\frac{1}{x}, 0 \leq y \leq x \leq 1\) (c) \(f_{X, Y}(x, y)=6 x, 0 \leq x \leq 1,0 \leq y \leq 1-x\)

Find and graph \(f_{X, Y}(x, y)\) if the joint cdf for random variables \(X\) and \(Y\) is $$ F_{X, Y}(x, y)=x y, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1 $$

Find the joint pdf associated with two random variables \(X\) and \(Y\) whose joint cdf is $$ F_{X, Y}(x, y)=\left(1-e^{-\lambda y}\right)\left(1-e^{-\lambda x}\right), \quad x>0, \quad y>0 $$

Prove that $$ \begin{aligned} P(a

A certain brand of fluorescent bulbs will last, on the average, one thousand hours. Suppose that four of these bulbs are installed in an office. What is the probability that all four are still functioning after one thousand fifty hours? If \(X_{i}\) denotes the \(i\) th bulb's life, assume that $$ f_{X_{1}, X_{2}, X_{3}, X_{4}}\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\prod_{i=1}^{4}\left(\frac{1}{1000}\right) e^{-x / 1000} $$

A hand of six cards is dealt from a standard poker deck. Let \(X\) denote the number of aces, \(Y\) the number of kings, and \(Z\) the number of queens. (a) Write a formula for \(p_{X, Y, Z}(x, y, z)\). (b) Find \(p_{X, Y}(x, y)\) and \(p_{X, Z}(x, z)\).

Calculate \(p_{X, Y}(0,1)\) if \(p_{X, Y, Z}(x, y, z)=\) \(\frac{3 !}{x[y ! z !(3-x-y-z) !}\left(\frac{1}{2}\right)^{x}\left(\frac{1}{12}\right)^{y}\left(\frac{1}{6}\right)^{z} \cdot\left(\frac{1}{4}\right)^{3-x-y-z}\) for \(x, y, z=0,1\), 2,3 and \(0 \leq x+y+z \leq 3\).

Suppose that the random variables \(X, Y\), and \(Z\) have the multivariate pdf $$ f_{X, Y, Z}(x, y, z)=(x+y) e^{-z} $$ for \(00\). Find (a) \(f_{X, Y}(x, y)\), (b) \(f_{Y, Z}(y, z)\), and (c) \(f_{Z}(z)\).

The four random variables \(W, X, Y\), and \(Z\) have the multivariate pdf $$ f_{W, X, Y, Z}(w, x, y, z)=16 w x y z $$ for \(0 \leq w \leq 1,0 \leq x \leq 1,0 \leq y \leq 1\), and \(0 \leq z \leq 1\). Find the marginal pdf, \(\bar{f}_{W, X}(w, x)\), and use it to compute \(P\left(0 \leq W \leq \frac{1}{2}, \frac{1}{2} \leq X \leq 1\right) .\)

Two fair dice are tossed. Let \(X\) denote the number appearing on the first die and \(Y\) the number on the second. Show that \(X\) and \(Y\) are independent.

Let \(f_{X, Y}(x, y)=\lambda^{2} e^{-\lambda(x+y)}, 0 \leq x, 0 \leq y .\) Show that \(X\) and \(Y\) are independent. What are the marginal pdfs in this case?

Suppose that each of two urns has four chips, numbered 1 through 4 . A chip is drawn from the first urn and bears the number \(X\). That chip is added to the second urn. A chip is then drawn from the second urn. Call its number \(Y\).

Let \(X\) and \(Y\) be random variables with joint pdf $$ f_{X, Y}(x, y)=k, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1, \quad 0 \leq x+y \leq 1 $$

Are the random variables \(X\) and \(Y\) independent if \(f_{X, Y}(x, y)=\frac{2}{3}(x+2 y), 0 \leq x \leq 1,0 \leq y \leq 1 ?\)

Suppose that random variables \(X\) and \(Y\) are independent with marginal pdfs \(f_{X}(x)=2 x, 0 \leq x \leq 1\), and \(f_{Y}(y)=3 y^{2}, 0 \leq y \leq 1\). Find \(P(Y

. Find the joint cdf of the independent random variables \(X\) and \(Y\), where \(f_{X}(x)=\frac{x}{2}, 0 \leq x \leq 2\), and \(f_{Y}(y)=2 y, 0 \leq y \leq 1 .\)

If two random variables \(X\) and \(Y\) are independent with marginal pdfs \(f_{X}(x)=2 x, 0 \leq x \leq 1\), and \(f_{Y}(y)=1\), \(0 \leq y \leq 1\), calculate \(P\left(\frac{Y}{X}>2\right)\).

Suppose \(f_{X, Y}(x, y)=x y e^{-(x+y)}, x>0, y>0\). Prove for any real numbers \(a, b, c\), and \(d\) that $$ P(a

Given the joint pdf \(f_{X, Y}(x, y)=2 x+y-2 x y\), \(0

Prove that if \(X\) and \(Y\) are two independent random variables, then \(U=g(X)\) and \(V=h(Y)\) are also independent.

If two random variables \(X\) and \(Y\) are defined over a region in the \(X Y\)-plane that is not a rectangle (possibly infinite) with sides parallel to the coordinate axes, can \(X\) and \(Y\) be independent?

Write down the joint probability density function for a random sample of size \(n\) drawn from the exponential pdf, \(f_{X}(x)=(1 / \lambda) e^{-x / \lambda}, x \geq 0 .\)

Suppose that \(X_{1}, X_{2}, X_{3}\), and \(X_{4}\) are independent random variables, each with pdf \(f_{X_{i}}\left(x_{i}\right)=4 x_{i}^{3}, 0 \leq x_{i} \leq 1\). Find (a) \(P\left(X_{1}<\frac{1}{2}\right)\). (b) \(P\left(\right.\) exactly one \(\left.X_{i}<\frac{1}{2}\right)\). (c) \(f_{X_{1}, X_{2}, X_{3}, X_{4}}\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\). (d) \(F_{X_{2}, X_{3}}\left(x_{2}, x_{3}\right)\).

A random sample of size \(n=2 k\) is taken from a uniform pdf defined over the unit interval. Calculate \(P\left(X_{1}<\frac{1}{2}, X_{2}>\frac{1}{2}, X_{3}<\frac{1}{2}, X_{4}>\frac{1}{2}, \ldots, X_{2 k}>\frac{1}{2}\right) .\)

Let \(Y\) be a continuous random variable with \(f_{Y}(y)=\frac{1}{2}(1+y),-1 \leq y \leq 1\). Define the random variable \(W\) by \(W=-4 Y+7\). Find \(f_{W}(w) .\) Be sure to specify those values of \(w\) for which \(f_{W}(w) \neq 0\).

Let \(f_{Y}(y)=\frac{3}{14}\left(1+y^{2}\right), 0 \leq y \leq 2\). Define the random variable \(W\) by \(W=3 Y+2\). Find \(f_{W}(w)\). Be sure to specify the values of \(w\) for which \(f_{W}(w) \neq 0\).

Let \(X\) and \(Y\) be two independent random variables. Given the marginal pdfs shown below, find the pdf of \(X+Y\). In each case, check to see if \(X+Y\) belongs to the same family of pdfs as do \(X\) and \(Y\). (a) \(p_{X}(k)=e^{-\lambda} \frac{\lambda^{k}}{k !}\) and \(p_{Y}(k)=e^{-\mu} \frac{\mu^{k}}{k !}, k=0,1,2, \ldots\) (b) \(p_{X}(k)=p_{Y}(k)=(1-p)^{k-1} p, k=1,2, \ldots\)

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

Let \(X\) and \(Y\) be two independent random variables, whose marginal pdfs are given below. Find the pdf of \(X+Y\). (Hint: Consider two cases, \(0 \leq w<1\) and \(1 \leq w \leq 2 .)\) $$ f_{X}(x)=1,0 \leq x \leq 1, \text { and } f_{Y}(y)=1,0 \leq y \leq 1 $$

If a random variable \(V\) is independent of two independent random variables \(X\) and \(Y\), prove that \(V\) is independent of \(X+Y\).