Exercise 1 (30 POINTS) Three balanced coins are tossed independently. One of the variables of interest is Y, the number of heads. Let Y denote the amount of money won on a side bet in the following manner. If the first head occurs on the first toss, you win $1. If the first head occurs on toss 2 or on toss 3 you win $2 or $3, respectively. If no heads appear, you lose $1 (that is, win -$1). a. Find the joint probability function for Y and Y. b. What is the probability that fewer than three heads will occur and you will win $1 or less? [That is, find F(2, 1).] c. Derive the marginal probability distribution for your winnings on the side bet. d. What is the probability that you obtained three heads, given that you won $1 on the side bet? e. The number of heads in three coin tosses is binomially distributed with,n = 3 and p = . Are the total number of heads and your winnings on the side bet independent? EXPLAIN!