Chapter 7.4, Problem 50E

Consider the random variables x_R and x_B defined in the previous exercise.

1. a. Suppose that you are offered a choice of the following two games:

   Game 1:    Pay $7 to play, and you win y₁, dollars, where $y_1=x_R+x_B$.

Game 2:    Doesn’t cost anything to play initially, but you “win” 3y₂ dollars, where $y_2=x_R-x_B$. If y₂ is negative, you must pay that amount; if it is positive, you receive that amount.

For Game 1, the net amount won in a game is $w_1=y_1-7=x_R+x_B-7$. What are the mean and standard deviation of w₁?

1. b. For Game 2, the net amount won in a game is $w_2-3y_2=3\left(x_R-x_B\right)$. What are the mean and standard deviation of w₂?
2. c. Based on your answers to Parts (a) and (b), if you had to play, which game would you choose? Explain your reasoning.

7.49. Consider a game in which a red die and a blue die are rolled. Let x_R denote the value showing on the uppermost face of the red die, and define x_B similarly for the blue die.

1. a. The probability distribution of x_R is

Find the mean, variance, and standard deviation of x_R.

1. b. What are the values of the mean, variance, and standard deviation of x_B? (You should be able to answer this question without doing any additional calculations.)