The number of orders for installation of a computer information system arriving at an agency per week is a random variable X with the following probability distribution:
x P(x)

0 .......0.10

1 .......0.20

2 .......0.30

3 .......0.15

4 .......0.15

5 .......0.05

6 .......0.05

a. Prove that P(X) is a probability distribution.
b. Find the cumulative distribution function of X.

c. Use the cumulative distribution function to find probabilities P(2 < X ≤ 5), P(3 ≤ X ≤ 6), and P(X > 4).

d. What is the probability that either four or five orders will arrive in a given week?

e. Assuming independence of weekly orders, what is the probability that three orders will arrive next week and the same number of orders the following week?

f. Find the mean and the standard deviation of the number of weekly orders.