The Bombay Company offers reproductions of classic 18th- and 19th-century English furniture pieces, which have become popular in recent years. The following table gives the probability distribution of the number of Raffles tables sold per day at a particular Bombay store.

Number of TablesProbability

0 ..........0.05

1 ..........0.05

2 ..........0.10

3 ..........0.15

4 ..........0.20

5 ..........0.15

6 ..........0.15

7 ..........0.10

8 ..........0.05

a. Show that the probabilities above form a proper probability distribution.
b. Find the cumulative distribution function of the number of Raffles tables sold daily.

c. Using the cumulative distribution function, find the probability that the number of tables sold in a given day will be at least three and less than seven.

d. Find the probability that at most five tables will be sold tomorrow.

e. What is the expected number of tables sold per day?

f. Find the variance and the standard deviation of the number of tables sold per day.

g. Use Chebyshev's theorem to determine bounds of at least 0.75 probabilities on the number of tables sold daily. Compare with the actual probability for these bounds using the distribution itself.