Question 3 Consider the following game you can play by payin

Question 3. Consider the following game you can play by paying $1: You draw one ticket at random from a box of numbered tickets. You win the amount (in dollars) of the number on the ticket you draw, which is denoted by the random variable X. Two boxes are available with the numbered tickets as shown below. Box 1 0 1 2 Box 2 0 0 0 1 4

(a) Define, Y , the random variable for your net gain, in terms of X. 1

(b) Find E[Y ] and V ar(Y ) per play with Box 1.

(c) Find E[Y ] and V ar(Y ) per play with Box 2.

(d) Given that you have decided to play, which box would you choose and why?

Solution

a)

As you play, you pay $1 first, your profit Y is

Y = X - 1 [ANSWER]

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b)

COnsider the table:

As

E[Y] = Sum(Y P(Y)) = 0
Var(Y) = E(Y^2) - E(Y)^2 = 0.66666667

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c)

b)

COnsider the table:

As

E[Y] = Sum(Y P(Y)) = 0
Var(Y) = E(Y^2) - E(Y)^2 = 2.4

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d) Both just have the same expected profit, which is 0. However, you are more certain of what will happen with box 1, as it has lesser variance. Thus, you might as well go for BOX 1. [ANSWER]