In the New York State Lottery game "Take 5," five numbers are chosen at random without replacement from the numbers 1,2,,39 . Let X denote the number of even numbers chosen. The random variable X has the following probability table:

If there are X even numbers chose, then the number of odd numbers chosen is 5 X . Let Y be the difference between the number of even and the number of odd numbers chosen, so that Y = X (5 X )= 2X 5 .

1. What is the probability that exactly one even number is chosen?

2. What is the probability that at most one even number is chosen?

3. What is the probability that there are more even than odd numbers chosen?

4. Show that the mean of X is E (X )= 2.4359 .

x	0	1	2	3	4	5
P (X = x )	0.02693	0.15989	0.33858	0.31977	0.13464	0.02020

5. Show that the variance of X is Var (X ) = 1.1177 .

6. What is the standard deviation of X ?

7. Find the mean of Y , i.e., find E (Y ).

8. Find the variance of Y , i.e., find Var (Y ).