Problem 1 (Discrete Random Variables): Define the random experiment: Roll two four-sided dice.

Define X = Minimum (Die 1, Die 2).

Define Y = |Die 1 – Die 2|.

a) Sketch the PDF of X.

b) Compute E(X).

c) Sketch the PDF of Y.

d) Compute E(Y).

Problem 2 (Expected Value and Variance): The following graph depicts the probability distribution function (PDF) for the random variable X.

a) Compute E(X).

b) Compute Var(X).

Var(X) = E(X – E(X)) 2

OR Var(X) = E(X2 ) – E(X)2

Problem 3 (Expected Value): Define the random experiment: Roll two four-sided dice. Define X = Minimum (Die 1, Die 2). How much would you be willing to wager ($w) that X is greater than 1? You will win $1 if you are correct and lose your wager if you are incorrect.

Problem 4 (Expected Value): You attend a summer fair event at which one of the booths is the game “Skee-Ball”. For $2, you can purchase 3 balls and have 3 opportunities to roll each ball into a centrally located target. You will receive your $2 back and win an additional $1 if you hit the target once, an additional $3 if you hit the target twice and an additional $5 if you hit the target three times. If you do not hit the target at all, you will lose your initial $2. Define the following random variables and their probability distributions.

Let X = Number of balls that hit the target for an average player:

Let W = Average Player’s Winnings

P(W = -2) = P(X = 0) = 0.512

P(W = +1) = P(X = 1) = 0.384

P(W = +3) = P(X = 2) = 0.096

P(W = +5) = P(X = 3) = 0.008

True or False, the game is profitable for an average player?