Applied Statistics HW 14

1. We have a 4-sided die where the side 1 is three times as likely as all the others.
   1. Denote by X the outcome of rolling the die once. What is the probability distribu- tion for X?
   2. What are the mean and standard deviation?
   3. Suppose we now roll the die twice. Denote by \(X_1\) the outcome of the first roll, by \(X_2\) the outcome of the second roll, and by \(Y\) the sum of the two outcomes. Describe how Y relates to the other two, and use this information to construct the probability table for Y.
   4. Compute the mean and standard deviation of Y both from the table directly, and using the formula on how Y relates to \(X_1\) and \(X_2\) and the mean and stand deviation values for those.
2. Peter has bought shares in two stocks. Let’s denote by X the profit that one share of stock A makes for him after a week, and by Y the profit that one share of stock B makes for him after the same week. We don’t of course know precisely what X,Y would be, but let’s imagine that X has a mean of $10 with a standard deviation of $3, while Y has a mean of $20 with a standard deviation of $5, and also that they are independent of each other. He has 15 shares of stock A and 25 shares of stock B. The overall profit he makes would then be \(Z = 15X + 25Y\). What are the mean and standard deviation for his overall profit?
3. Alice and Bob play the following game: They roll a 6-sided die ONCE. Alice earns a dollar as long as the die is 4 or more. Bob earns two dollars as long as the die is a 1 or a 6 (on that same roll). So it is possible that they both benefit from the roll, or only one of them, or neither. Let’s denote by X the difference in dollars earned between Alice and Bob, namely how much Alice made minus how much Bob made.
   1. What is the probability distribution of X?
   2. What are the mean and standard deviation of X?