Greedy Bob Game

An interesting game problem asked Senior Quant Interview in Millenium

Alice and Bob are playing a game. Alice spins a spinner and wins X dollars, with mean 12 and variance 2. Bob flips a coin, and if the coin lands on heads, then Bob takes from Alice whatever she won, else Bob wins nothing. What is Var(Y), where Y denotes the amount Bob wins?

Try to solve this problem yourself before moving on to the solution below

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Solution

Let I of mean 0.5 and variance 0.25 be the indicator variable that denotes if Bob gets a head. Then,

\(Y = I \times X \)

, where I and X are independent. Thus we can write:

\( \text{Var}(Y) = \text{Var}(I \times X) = (\text{Var}(I) + \text{E}[I]^2)(\text{Var}(X) + \text{E}[X]^2) - \text{E}[I]^2 \text{E}[X]^2 \)

\(\\ Var(Y) = (0.25 + 0.25)(2 + 144) - (0.25 \times 144) = 37 \)

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😃 Over to you: Were you able to solve this?

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