Question

Problem 8.2

In a simple gambling game, you roll a single fair 6-sided die until the first time you get a 1. After the first 1 appears, your winnings W in dollars is the total number of points on all your rolls (including the last one).

(a) Define random variables N, X₁, X₂, ... such that W = X₁ + X₂ + ···+ XN . (The answer is not unique, but we can probably all agree on the most useful random variables for modeling the situation.)

(b) Are the random variables in (a) mutually independent?

(c) Find E[W] and Var[W]. Hint: If you found that your random variables in (a) are not independent, be careful to not use a formula that only applies when N, X₁, X₂, ... are mutually independent.

(d) Show that

_{(Simple version: Find a single value of n є {1, 2, . ·.) for which the two expressions are not equal. More fun: Evaluate each expression as a function of n.)}

Answer 1

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