# Calculating Expected Value and Variance Given Random Variable Distributions.

I am trying to solve the following question from one of my classes.

I generally understand how to find the expected value and variance when given a random variable. However, when doing so, it is usually with an example where the values and probabilities of the random value are pretty easily defined (like rolling a die). In this example, I'm unsure of how to go about setting up and solving the problem because I'm not sure what values X or Y could take on and what the probabilities of those values would be. Could anyone show how to find expected values and variances in this problem?

The question tells you exactly what values $X$ and $Y$ can take and what the probabilities are. $X$ takes the values $2$ and $-2$, each with probability $1/2$. $Y$ takes the values $4$ and $-1$, with probabilities $0.2$ and $0.8$.
• Your calculation of $E(X^2)$ is wrong: $(-2)^2 = 2^2$, not $-(2^2)$. Also notice that $(100 X)^2 = 100^2 X^2$, not $100 X^2$. – Robert Israel Sep 22 '14 at 4:47
"I will throw a fair coin, if it comes up heads I will give you $2$ dollars if not, you will give me $2$ dollars. You can also choose to play with unfair coin which comes up heads $20 \%$ of the time. If it comes up heads I will give you $4$ dollars if not, you will give me $1$ dollar. "
In the fair coin case your expected profit: $$2*0.5+(-2)*0.5 = 0$$ In the unfair coin case: $$4*0.2+(-1)*0.8=0$$
It is better not to waste your time with him because your expected profit is $0$.