e. Your friend will pay you $4$ dollars if you win the game. You owe your friend $1$ dollar if you lose the game. What are your expected winnings?
f. What is the variance of your expected winnings?
My work:
For e) I know that $E(X) = p = \cfrac{1}{6}$ Using this information, I can calculate the expected winnings $E(X) = 4 \cdot \cfrac{1}{6} + -1 \cdot \cfrac{5}{6} = \cfrac{-1}{6}$ is the expected winnings
f) I need help with this one. I know for a Bernoulli random variable, $Var(X) = pq = \cfrac{1}{6} \cdot \cfrac{5}{6} = \cfrac{5}{36}$ But how can I use this information to calculate the variance of the expected winnings?
Can someone give me a clue?
Thank You!
Calculate $E(X^2)$ as $4^2\cdot\frac16+(-1)^2\cdot\frac56=\frac{21}6$. Then the variance is $E(X^2)-E(X)^2=\frac{21}6-\left(-\frac16\right)^2=\frac{125}{36}$.