Suppose outcome space $\Omega = {1,2,3,4,5,6}$ represents a 6-sided die, and the probability function assigns probability $1/6$ to each outcome. Let random variable $X: \Omega \rightarrow \mathbb{R}$ be defined as, $$X(\omega) = \begin{cases} 1, &\omega \le 2\\ 2, &otherwise\end{cases}$$ Compute the pmf of $X$

What value of $c$ makes the following a valid probability mass function? $$f(x) = \begin{cases} c, &x \in {1,2}\\ 2c, &x \in {5}\\ 0, &otherwise \end{cases}$$

Suppose $X$ has density function $f(x)=c(3−|x|)$ when $−3<x<3$. What value of $c$ makes this a density function?

Consider $f(x)=c(1−x^2)$ for $−1<x<1$ and $0$ otherwise. What value of $c$ should we take to make $f$ a density function?

Let X :== as a Random Variable representing the number of heads in four independent flips of a fair coin.

1. Provide in bracket notation the probability mass function (pmf) of X;
2. Compute the probability that X is equal to an odd number;
3. Provide in bracket notion the cumulative distribution function (cdf) of X
4. What are the probailities P(X <= 2), P(X <= 1.75) and P(X >= 2.3)

Problem modified afterHogg, McKean and Craig - Introduction to Mathematical Statistics

Suppose $X$ has density function $f(x)=x/2$ for $0<x<2$ and $0$ otherwise. Find

(a) the cumulative distribution function,

(b) $P(X<1)$,

(c) $P(X>3/2)$

Suppose the CDf of X is $F$, where $F(x)=e^{−1/x}$ for $x>0$ and $F(x)=0$ for $x\leq0$. Is $X$ a continuous random variable?

Let $F(x)=3x-2x^2$ for $0\leq x\leq1$, $F(x)=0$ for $x\leq0$, and $F(x)=1$ for $x\geq1$. Is $F$ a valid cumulative distribution function?

Suppose we roll two dice and let $X$ and $Y$ be the two numbers that appear. Find the distribution of $|X-Y|$.

Suppose $X$ has density function $f(x)$ for $a\leq x\leq b$ and $Y=cX+d$, where $c>0$. Find the density function of $Y$.

Compute (a) $P(X=1|Y=1)$ and (b) $P(X = 2|Y=2)$ for the following joint distribution:

Using the clues given below, fill in the rest of the joint distribution. There is only one answer:

For $k=1, 2, 3$, (a) $P(Y=1|X=k)=\frac{2}{3}$, (b) $P(X=k|Y=1)=\frac{k}{6}$.

Let X and Y be jointly continuous random variables with joint distribution shown below: $f_{XY}(X,Y)= \begin{cases} cx+1 & \text{if } x,y \geq 0, x+y < 1\ 0 & \text{otherwise} \end{cases}$ 1) Find the constant c. 2) Are X and Y independent random variables?

Fill in the rest of the joint distribution so that $X$ and $Y$ are independent. There are two possible answers:

You are excited about a concert featuring your favorite a cappella group: the Pitch Estimators. Tickets go on sale at noon, but before you can buy a ticket, you have to wait for your turn in an online waiting room. Because tickets are in high demand, you enlist two of your friends to help you. All three of you enter the waiting room at noon, and as soon as any one of you gets a ticket, you are done and can all sign off.

Suppose your waiting time in minutes is a continuous random variable $T$. Your first friend's waiting time in minutes is a continuous random variable $U$. Your second friend's waiting time in minutes is a continuous random variable $V$.

Suppose these random variables are are mutually independent and have probability density functions given by:

$$f_T(t)=\begin{cases} \frac{1}{3}e^{-\frac{1}{3}t}, &t \geq 0\ 0, &otherwise \end{cases}$$

$$f_U(t)=\begin{cases} \frac{1}{2}e^{-\frac{1}{2}t}, &t \geq0\ 0, &otherwise \end{cases}$$

$$f_V(t)=\begin{cases} \frac{1}{6}e^{-\frac{1}{6}t}, &t\geq 0\ 0, &otherwise \end{cases}$$

These are examples of what we call exponential random variables.

a. Please sketch the probability density for all three random variables on one graph.

b. For a particular time $t$, compute the probability that $T>t$, $U>t$, and $V>t$.

c. Let $O$ represent your overall waiting time. $O$ is the minimum of the waiting times for each of the three websites, $O = min(T,U,V)$. Compute the probability density function of $O$. (Hint: use the previous answer to write down the cdf)

The following statements are either true or false. Prove them or provide a counterexample:

1. If X, Y, and Z are random variables, X and Y are independent, Y and Z are independent, then X and Z must be independent.

2. If X, Y, and Z are random variables, X and Y are not independent, Y and Z are not independent, then X and Z must be not independent.