Q2. (a) The following game is offered. There are 10 cards face-down numbered 1 through

10. You can pick one card. Your payoff is $0.50 if the number on the card is less than

5 and is the dollar value on the card otherwise. What are the expected value and the

standard deviation of your payoff? [5]

(b) Show that Binomial random variable is the summation of Bernoulli random

variables. [3]

(c) You have two fair coins. The first coin is tossed five times. Let the random variable

X be the number of heads showing up in these five tosses. The second coin is tossed X times. Let Y be the number of heads showing up in the tosses of the second coin.

What is the joint probability mass function of X and Y ? What is E(X + Y )?