Variance of a coin-gated share of a spinner prize

Question

Alice spins a wheel and wins $X$ dollars, where $\mathbb{E}X=12$ and $\operatorname{Var}X=2$ . Bob then tosses a fair coin independently: on heads he takes all of Alice's winnings, on tails he gets nothing. Let $Y$ be Bob's take. Find $\operatorname{Var}(Y)$ .

Short answer

$37$ . $Y=IX$ with $I$ an independent fair-coin indicator. Then $\mathbb{E}Y=6$ and $\mathbb{E}Y^2=\mathbb{E}I\cdot\mathbb{E}X^2=\tfrac12(2+144)=73$ , so $\operatorname{Var}Y=73-36=37$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Covariance and Correlation
Populations, Samples, and Sampling Distributions

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free