Homework 3

1. If you play a game where you

   • Win $10 with probability 0.2

   • Win $6 with probability 0.15

   • Win $2 with probability 0.3

   • Lose $5 with probability 0.1

   • Lose $10 with probability 0.25

What is the expected amount of money won/lost if each of the possible win/loss amounts is squared?

2. If X is a continuous RV with pdf f(x) = 5x^4 on [0, 1]. What is E(X^4)?

3. If X is the RV from #2, what is the median of X?

4. If, in one year, an investment account has a

   • 19% chance of doubling (2x)

   • 20% chance of halving (0.5x)

   • 41% chance of growing 15% (1.15x)

   • 20% chance of shrinking 5% (0.95x)

What is the average yearly production rate of this account?

5. Compute the variance of the game in #1.

6. Compute the variance of X in #2

7. Meals are randomly priced in your city. Suppose the average meal costs $10 with a variance of $2. What is the most the probability can be that a random meal is more than $6 from the mean (either 16 or more or 4 or less)?

8. Suppose you play two games where one influences another. Suppose the win probabilities are as follows:

   • You have a 9% chance win both game 1 and game 2

   • You have a 40% chance to lose both game 1 and game 2.

   • You have a 17% chance to win game 1.

Write games 1 and 2 as separate Bernoulli RVs and fill in the rest of the joint distribution table for games 1 and 2.

9. In #8, suppose the two games have the following payouts:

   • Game 1: $8 for winning, -$1 for losing

   • Game 2: $5 for winning,-$6 for losing.

Find the average total payout from playing 1 round of each game.

10. In #8, find the covariance of the two games.