Question

A fair four-sided die has two faces numbered 0 and two faces numbered 2. Another fair four-sided die has its faces numbered 0, 1, 4, and 5. The two dice are rolled. Let X and Y be the respective outcomes of the roll. Let W = X + Y. a. Determine the pmf of W. b. Draw a probability histogram of the pmf of W.

Answer 1

The two dice have PMFs

$P(X=x)=\begin{cases}\dfrac12&\text{for }x\in\{0,2\}\\\\0&\text{otherwise}\end{cases}$

$P(Y=y)=\begin{cases}\dfrac14&\text{for }y\in\{0,1,4,5\}\\\\0&\text{otherwise}\end{cases}$

The sum of the two rolls can take on 8 different values:

$W=\begin{cases}0&\text{if }x=0,y=0\\1&\text{if }x=0,y=1\\2&\text{if }x=2,y=0\\3&\text{if }x=2,y=1\\4&\text{if }x=0,y=4\\5&\text{if }x=0,y=5\\6&\text{if }x=2,y=4\\7&\text{if }x=2,y=5\end{cases}$

The outcome of either roll is independent of the other, so that

$P(X=x\text{ and }Y=y)=P(X=x)P(Y=y)$

a. Then PMF of $W$ is

$P(W=w)=\begin{cases}\dfrac12\cdot\dfrac14=\dfrac18&\text{for }w\in\{0,1,2,3,4,5,6,7\}\\\\0&\text{otherwise}\end{cases}$

b. The histogram is nothing special, same as the discrete uniform distribution over the interval [0, 7].