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Problem
Consider two random variables $X$ and $Y$ with joint PMF given in Table 5.4

Joint PMF of $X$ and $Y$ in Problem 1

	$Y = 1$	$Y = 2$
$X = 1$	$\frac{1}{3}$	$\frac{1}{12}$
$X = 2$	$\frac{1}{6}$	$0$
$X = 4$	$\frac{1}{12}$	$\frac{1}{3}$

1. Find $P(X \leq 2, Y > 1)$.
2. Find the marginal PMFs of $X$ and $Y$.
3. Find $P(Y=2 | X=1)$.
4. Are $X$ and $Y$ independent?

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Problem
Let $X$ and $Y$ be as defined in Problem 1. I define a new random variable $Z=X-2Y$.

1. Find the PMF of $Z$.
2. Find $P(X=2 | Z=0)$.

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Problem
A box contains two coins: a regular coin and a biased coin with $P(H)=\frac{2}{3}$. I choose a coin at random and toss it once. I define the random variable $X$ as a Bernoulli random variable associated with this coin toss, i.e., $X=1$ if the result of the coin toss is heads and $X=0$ otherwise. Then I take the remaining coin in the box and toss it once. I define the random variable $Y$ as a Bernoulli random variable associated with the second coin toss. Find the joint PMF of $X$ and $Y$. Are $X$ and $Y$ independent?

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Problem
Consider two random variables $X$ and $Y$ with joint PMF given by \begin{align}%\label{} P_{XY}(k,l)=\frac{1}{2^{k+l}}, \textrm{for }k,l=1,2,3,... \end{align}

1. Show that $X$ and $Y$ are independent and find the marginal PMFs of $X$ and $Y$.
2. Find $P(X^2+Y^2 \leq 10)$.

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Problem
Let $X$ and $Y$ be as defined in Problem 1. Also, suppose that we are given that $Y=1$.

1. Find the conditional PMF of $X$ given $Y=1$. That is, find $P_{X|Y}(x|1)$.
2. Find $E[X|Y=1]$.
3. Find $Var(X|Y=1)$.

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Problem
The number of customers visiting a store in one hour has a Poisson distribution with mean $\lambda=10$. Each customer is a female with probability $p=\frac{3}{4}$ independent of other customers. Let $X$ be the total number of customers in a one-hour interval and $Y$ be the total number of female customers in the same interval. Find the joint PMF of $X$ and $Y$.

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Problem
Let $X \sim Geometric(p)$. Find $Var(X)$ as follows: Find $EX$ and $EX^2$ by conditioning on the result of the first "coin toss", and use $Var(X)$$=EX^2-(EX)^2$.

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Problem
Let $X$ and $Y$ be two independent $Geometric(p)$ random variables. Find $E\left[\frac{X^2+Y^2}{XY}\right]$.

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Problem
Consider the set of points in the set $C$: \begin{align}%\label{} \nonumber C=\{(x,y) | x,y \in \mathbb{Z}, x^2+ |y| \leq 2\}. \end{align} Suppose that we pick a point $(X,Y)$ from this set completely at random. Thus, each point has a probability of $\frac{1}{11}$ of being chosen.

1. Find the joint and marginal PMFs of $X$ and $Y$.
2. Find the conditional PMF of $X$ given $Y=1$.
3. Are $X$ and $Y$ independent?
4. Find $E[XY^2]$.

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Problem
Consider the set of points in the set $C$: \begin{align}%\label{} \nonumber C=\{(x,y) | x,y \in \mathbb{Z}, x^2+ |y| \leq 2\}. \end{align} Suppose that we pick a point $(X,Y)$ from this set completely at random. Thus, each point has a probability of $\frac{1}{11}$ of being chosen.

1. Find $E[X|Y=1]$.
2. Find $Var(X|Y=1)$.
3. Find $E[X| |Y| \leq 1]$.
4. Find $E[X^2 | |Y| \leq 1]$.

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Problem
The number of cars being repaired at a small repair shop has the following PMF: \begin{equation} \nonumber P_N(n) = \left\{ \begin{array}{l l} \frac{1}{8} & \quad \text{for } n=0\\ \frac{1}{8} & \quad \text{for } n=1\\ \frac{1}{4} & \quad \text{for } n=2\\ \frac{1}{2} & \quad \text{for } n=3\\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation} Each car that is being repaired is a four-door car with probability $\frac{3}{4}$ and a two-door car with probability $\frac{1}{4}$, independently from other cars and independently from the number of cars being repaired. Let $X$ be the number of four-door cars and $Y$ be the number of two-door cars currently being repaired.

1. Find the marginal PMFs of $X$ and $Y$.
2. Find the joint PMF of $X$ and $Y$.
3. Are $X$ and $Y$ independent?

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Problem
Let $X$ and $Y$ be two independent random variables with PMFs \begin{equation} \nonumber P_X(k)=P_Y(k) = \left\{ \begin{array}{l l} \frac{1}{5} & \quad \text{for } x=1,2,3,4,5\\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation} Define $Z=X-Y$. Find the PMF of $Z$.

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Problem
Consider two random variables $X$ and $Y$ with joint PMF given in Table 5.5

Table 5.5: Joint PMF of X and Y in Problem 13

	$Y = 0$	$Y = 1$	$Y = 2$
$X = 0$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{8}$
$X = 1$	$\frac{1}{8}$	$\frac{1}{6}$	$\frac{1}{4}$

Define the random variable $Z$ as $Z=E[X|Y]$.

1. Find the Marginal PMFs of $X$ and $Y$.
2. Find the conditional PMF of $X$, given $Y=0$ and $Y=1$, i.e., find $P_{X|Y}(x|0)$ and $P_{X|Y}(x|1)$.
3. Find the PMF of $Z$.
4. Find $EZ$, and check that $EZ=EX$.
5. Find Var$(Z)$.

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Problem
Let $X$, $Y$, and $Z=E[X|Y]$ be as in Problem 13. Define the random variable $V$ as $V=Var(X|Y)$.

1. Find the PMF of $V$.
2. Find $EV$.
3. Check that $Var(X)=EV+Var(Z)$.

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Problem
Let $N$ be the number of phone calls made by the customers of a phone company in a given hour. Suppose that $N \sim Poisson (\beta)$, where $\beta>0$ is known. Let $X_i$ be the length of the $i$'th phone call, for $i=1,2,..., N$. We assume $X_i$'s are independent of each other and also independent of $N$. We further assume \begin{align}%\label{} \nonumber X_i \sim Exponential(\lambda), \end{align} where $\lambda>0$ is known. Let $Y$ be the sum of the lengths of the phone calls, i.e., \begin{align}%\label{} \nonumber Y=\sum_{i=1}^{N}X_i. \end{align} Find $EY$ and Var$(Y)$.

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Problem
Let $X$ and $Y$ be two jointly continuous random variables with joint PDF \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} \frac{1}{2} e^{-x}+\frac{cy}{(1+x)^2} & \quad 0 \leq x, 0 \leq y \leq 1 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation}

1. Find the constant $c$.
2. Find $P(0 \leq X \leq 1, 0 \leq Y \leq \frac{1}{2})$.
3. Find $P(0 \leq X \leq 1)$.

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Problem
Let $X$ and $Y$ be two jointly continuous random variables with joint PDF \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} e^{-xy} & \quad 1 \leq x \leq e, y>0 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation}

1. Find the marginal PDFs, $f_X(x)$ and $f_Y(y)$.
2. Write an integral to compute $P(0 \leq Y \leq 1, 1 \leq X \leq \sqrt{e})$.

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Problem
Let $X$ and $Y$ be two jointly continuous random variables with joint PDF \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} \frac{1}{4}x^2+\frac{1}{6}y & \quad -1 \leq x \leq 1, 0 \leq y \leq 2 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation}

1. Find the marginal PDFs, $f_X(x)$ and $f_Y(y)$.
2. Find $P( X >0, Y<1)$.
3. Find $P( X >0 \textrm{ or } Y<1)$.
4. Find $P( X >0 | Y<1)$.
5. Find $P(X+Y>0)$.

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Problem
Let $X$ and $Y$ be two jointly continuous random variables with joint CDF \begin{equation} \nonumber F_{XY}(x,y) = \left\{ \begin{array}{l l} 1-e^{-x}-e^{-2y}+e^{-(x+2y)} & \quad x,y>0 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation}

1. Find the joint PDF, $f_{XY}(x,y)$.
2. Find $P(X<2Y)$.
3. Are $X$ and $Y$ independent?

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Problem
Let $X \sim N(0,1)$.

1. Find the conditional PDF and CDF of $X$ given $X>0$.
2. Find $E[X|X>0]$.
3. Find Var$(X|X>0)$.

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Problem
Let $X$ and $Y$ be two jointly continuous random variables with joint PDF \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} x^2+\frac{1}{3}y & \quad -1 \leq x \leq 1, 0 \leq y \leq 1 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation} For $0 \leq y \leq 1$, find the following:

1. The conditional PDF of $X$ given $Y=y$.
2. $P(X>0|Y=y)$. Does this value depend on $y$?
3. Are $X$ and $Y$ independent?

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Problem
Let $X$ and $Y$ be two jointly continuous random variables with joint PDF \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} \frac{1}{2}x^2+\frac{2}{3}y & \quad -1 \leq x \leq 1, 0 \leq y \leq 1 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation} Find $E[Y|X=0]$ and Var$(Y|X=0)$.

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Problem
Consider the set \begin{align}%\label{} \nonumber E=\{(x,y)| |x|+|y| \leq 1\}. \end{align} Suppose that we choose a point $(X,Y)$ uniformly at random in $E$. That is, the joint PDF of $X$ and $Y$ is given by \begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} c & \quad (x,y) \in E \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation}

1. Find the constant $c$.
2. Find the marginal PDFs $f_X(x)$ and $f_Y(y)$.
3. Find the conditional PDF of $X$ given $Y=y$, where $-1 \leq y \leq 1$.
4. Are $X$ and $Y$ independent?

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Problem
Let $X$ and $Y$ be two independent $Uniform(0,2)$ random variables. Find $P(XY<1)$.

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Problem
Suppose $X \sim Exponential(1)$ and given $X=x$, $Y$ is a uniform random variable in $[0,x]$, i.e., \begin{align}%\label{} \nonumber Y|X=x \hspace{10pt} \sim \hspace{10pt} Uniform(0,x), \end{align} or equivalently \begin{align}%\label{} \nonumber Y|X \hspace{10pt} \sim \hspace{10pt} Uniform(0,X). \end{align}

1. Find $EY$.
2. Find $Var(Y)$.

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Problem
Let $X$ and $Y$ be two independent $Uniform(0,1)$ random variables. Find

1. $E[XY]$
2. $E[e^{X+Y}]$
3. $E[X^2+Y^2+XY]$
4. $E[Ye^{XY}]$

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Problem
Let $X$ and $Y$ be two independent $Uniform(0,1)$ random variables, and $Z=\frac{X}{Y}$. Find the CDF and PDF of $Z$.

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Problem
Let $X$ and $Y$ be two independent $N(0,1)$ random variables, and $U=X+Y$.

1. Find the conditional PDF of $U$ given $X=x$, $f_{U|X}(u|x)$.
2. Find the PDF of $U$, $f_{U}(u)$.
3. Find the conditional PDF of $X$ given $U=u$, $f_{X|U}(x|u)$.
4. Find $E[X|U=u]$, and $Var(X|U=u)$.

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Problem
Let $X$ and $Y$ be two independent standard normal random variables. Consider the point $(X,Y)$ in the $x-y$ plane. Let $(R,\Theta)$ be the corresponding polar coordinates as shown in Figure 5.11. The inverse transformation is given by \begin{equation} \nonumber \left\{ \begin{array}{l} X=R \cos \Theta \\ Y=R \sin \Theta \end{array} \right. \end{equation} where, $R \geq 0$ and $-\pi < \Theta \leq \pi$. Find the joint PDF of $R$ and $\Theta$. Show that $R$ and $\Theta$ are independent.

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Problem
In Problem 29, suppose that $X$ and $Y$ are independent $Uniform(0,1)$ random variables. Find the joint PDF of $R$ and $\Theta$. Are $R$ and $\Theta$ independent?

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Problem
Consider two random variables $X$ and $Y$ with joint PMF given in Table 5.6.

Table 5.6: Joint PMF of X and Y in Problem 31

	$Y = 0$	$Y = 1$	$Y = 2$
$X = 0$	$\frac{1}{6}$	$\frac{1}{4}$	$\frac{1}{8}$
$X = 1$	$\frac{1}{8}$	$\frac{1}{6}$	$\frac{1}{6}$

Find $Cov(X,Y)$ and $\rho(X,Y)$.

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Problem
Let $X$ and $Y$ be two independent $N(0,1)$ random variable and \begin{align}%\label{} \nonumber &Z=11-X+X^2Y, \\ \nonumber &W=3-Y. \end{align} Find Cov$(Z,W)$.

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Problem
Let $X$ and $Y$ be two random variables. Suppose that $\sigma^2_X=4$, and $\sigma^2_Y=9$. If we know that the two random variables $Z=2X-Y $ and $W=X+Y$ are independent, find $Cov(X,Y)$ and $\rho(X,Y)$.

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Problem
Let $X \sim Uniform(1,3)$ and $Y|X \sim Exponential(X)$. Find $Cov(X,Y)$.

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Problem
Let $X$ and $Y$ be two independent $N(0,1)$ random variable and \begin{align}%\label{} \nonumber &Z=7+X+Y, \\ \nonumber &W=1+Y. \end{align} Find $\rho(Z,W)$.

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Problem
Let $X$ and $Y$ be jointly normal random variables with parameters $\mu_X=-1$, $\sigma^2_X=4$, $\mu_Y=1$, $\sigma^2_Y=1$, and $\rho=-\frac{1}{2}$.

1. Find $P(X+2Y \leq 3)$.
2. Find $Cov(X-Y,X+2Y)$.

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Problem
Let $X$ and $Y$ be jointly normal random variables with parameters $\mu_X=1$, $\sigma^2_X=4$, $\mu_Y=1$, $\sigma^2_Y=1$, and $\rho=0$.

1. Find $P(X+2Y >4)$.
2. Find $E[X^2Y^2]$.

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Problem
Let $X$ and $Y$ be jointly normal random variables with parameters $\mu_X=2$, $\sigma^2_X=4$, $\mu_Y=1$, $\sigma^2_Y=9$, and $\rho=-\frac{1}{2}$.

1. Find $E[Y|X=3]$.
2. Find $Var(Y|X=2)$.
3. Find $P(X+2Y \leq 5 | X+Y=3)$.