Given the probability mass function P(x=0) = 1/3, P(x=1) = 1/8, and P(x=2) = 1/2, find E[x] and Var[x].

Here's the solution to your probability mass function problem.

The probability mass function (PMF) for the random variable $x$ is given as:
$P(x=0) = \frac{1}{3}$
$P(x=1) = \frac{1}{8}$
$P(x=2) = \frac{1}{2}$

First, let's check if the probabilities sum to 1:
$\frac{1}{3} + \frac{1}{8} + \frac{1}{2} = \frac{8}{24} + \frac{3}{24} + \frac{12}{24} = \frac{8+3+12}{24} = \frac{23}{24}$.
It seems there might be a slight error in the problem statement as the probabilities do not sum to 1. Assuming the problem intends for these to be the given probabilities, we will proceed with them. If the sum was intended to be 1, one of the probabilities would need to be adjusted. For example, if $P(x=2)$ was $\frac{13}{24}$ instead of $\frac{1}{2}$, the sum would be 1. However, we will use the values as written.

I) Find $E[x]$ and $Var[x]$