1. Consider a continuous random variable X with the following pdf:
where c is an unknown constant.
(a) What value of c makes f a valid pdf?
(b) What is E [X]?
(c) What is Var (X)? (Hint: first think about which variance formula is easiest to apply.)
(d) What is E[2X + 3]?
(e) What is Var(2X + 3)?
2. You flip a coin until you get a total of 3 heads. What is the expected number of flips this will take?
3. Let X ∼ Unif (-π, 0). What is E[sin(X)]?
4. You are playing a game where you roll a die and win 0 jellybeans for rolling a one or two; 1 jellybean for rolling a three, four, or five; and 2 jellybeans for rolling a six. Each time you play the game, you must pay 1 jellybean.
(a) What is the expected number of jellybeans you win each round?
(b) What is the variance of the number of jellybeans that you win?
(c) If you like jellybeans, is this a game you want to play?