Random Variables and Probability Distributions

BASIC

1. Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully.

3. A discrete random variable X has the following probability distribution:

$\begin{align*} \begin{array}{c|ccccc} x & 77 & 78 & 79 & 80 & 81 \\ \hline P(x) & 0.15 & 0.15 & 0.20 & 0.40 & 0.10 \end{array} \end{align*}$

Compute each of the following quantities.

5. If each die in a pair is "loaded" so that one comes up half as often as it should, six comes up half again as often as it should, and the probabilities of the other faces are unaltered, then the probability distribution for the sum $X$ of the number of dots on the top faces when the two are rolled is

$\begin{aligned} &\begin{array}{c|cccccc} x & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline P(x) & \frac{1}{144} & \frac{4}{144} & \frac{8}{144} & \frac{12}{144} & \frac{16}{144} & \frac{22}{144} \end{array}\\ &\begin{array}{c|ccccc} x & 8 & 9 & 10 & 11 & 12 \\ \hline P(x) & \frac{24}{144} & \frac{20}{144} & \frac{16}{144} & \frac{12}{144} & \frac{9}{144} \end{array} \end{aligned}$

Compute each of the following.

APPLICATIONS

7. In a hamster breeder's experience the number $X$ of live pups in a litter of a female not over twelve months in age who has not borne a litter in the past six weeks has the probability distribution

$\begin{align*} \begin{array}{c|ccccccc} x & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline P(x) & 0.04 & 0.10 & 0.26 & 0.31 & 0.22 & 0.05 & 0.02 \end{array} \end{align*}$

9. Let $X$ denote the number of boys in a randomly selected three-child family. Assuming that boys and girls are equally likely, construct the probability distribution of $X$.

11. Five thousand lottery tickets are sold for $1 each. One ticket will win $1,000, two tickets will win $500 each, and ten tickets will win $100 each. Let $X$ denote the net gain from the purchase of a randomly selected ticket.

13. An insurance company will sell a $90,000 one-year term life insurance policy to an individual in a particular risk group for a premium of $478. Find the expected value to the company of a single policy if a person in this risk group has a 99.62% chance of surviving one year.

15. An insurance company estimates that the probability that an individual in a particular risk group will survive one year is 0.9825. Such a person wishes to buy a $150,000 one-year term life insurance policy. Let $C$ denote how much the insurance company charges such a person for such a policy.

17. A roulette wheel has 38 slots. Thirty-six slots are numbered from 1 to 36 ; half of them are red and half are black. The remaining two slots are numbered 0 and 00 and are green. In a $1 bet on red, the bettor pays $1 to play. If the ball lands in a red slot, he receives back the dollar he bet plus an additional dollar. If the ball does not land on red he loses his dollar. Let $X$ denote the net gain to the bettor on one play of the game.

19. The time, to the nearest whole minute, that a city bus takes to go from one end of its route to the other has the probability distribution shown. As sometimes happens with probabilities computed as empirical relative frequencies, probabilities in the table add up only to a value other than 1.00 because of round-off error.

$\begin{align*} \begin{array}{c|cccccc} x & 42 & 43 & 44 & 45 & 46 & 47 \\ \hline P(x) & 0.10 & 0.23 & 0.34 & 0.25 & 0.05 & 0.02 \end{array} \end{align*}$

ADDITIONAL EXERCISES

21. The number $X$ of nails in a randomly selected 1-pound box has the probability distribution shown. Find the average number of nails per pound.

$\begin{align*} \begin{array}{c|ccc} x & 100 & 101 & 102 \\ \hline P(x) & 0.01 & 0.96 & 0.03 \end{array} \end{align*}$

23. Two fair dice are rolled at once. Let $X$ denote the difference in the number of dots that appear on the top faces of the two dice. Thus for example if a one and a five are rolled, $X=4$, and if two sixes are rolled, $X=$ $0 .$

25. A manufacturer receives a certain component from a supplier in shipments of 100 units. Two units in each shipment are selected at random and tested. If either one of the units is defective the shipment is rejected. Suppose a shipment has 5 defective units.

27. The owner of a proposed outdoor theater must decide whether to include a cover that will allow shows to be performed in all weather conditions. Based on projected audience sizes and weather conditions, the probability distribution for the revenue $X$ per night if the cover is not installed is

The additional cost of the cover is $410,000. The owner will have it built if this cost can be recovered from the increased revenue the cover affords in the first ten 90-night seasons.