## Random Variables and Probability Distributions

This section first defines discrete and continuous random variables. Then, it introduces the distributions for discrete random variables. It also talks about the mean and variance calculations.

##### BASIC

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

a.

\begin{align*} \begin{array}{c|cccc} x & -2 & 0 & 2 & 4 \\ \hline P(x) & 0.3 & 0.5 & 0.2 & 0.1 \end{array} \end{align*}

b.

\begin{align*} \begin{array}{c|ccc} x & 0.5 & 0.25 & 0.25 \\ \hline P(x) & -0.4 & 0.6 & 0.8 \end{array} \end{align*}

c.

\begin{align*}\begin{array}{c|ccccc} x & 1.1 & 2.5 & 4.1 & 4.6 & 5.3 \\ \hline P(x) & 0.16 & 0.14 & 0.11 & 0.27 & 0.22 \end{array}\end{align*}

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.

a. $P(80)$.

b. $P(X>80)$.

c. $P(X \leq 80)$.

d. The mean $\mu$ of $X$.

e. The variance $\sigma^{2}$ of $X$.

f. The standard deviation $\sigma$ of $X$.

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.

a. $P(5 \leq X \leq 9)$.

b. $P(X \geq 7)$.

c. The mean $\mu$ of $X$. (For fair dice this number is 7 .)

d. The standard deviation $\sigma$ of $X$. (For fair dice this number is about $2.415$.)

##### 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*}

a. Find the probability that the next litter will produce five to seven live pups.

b. Find the probability that the next litter will produce at least six live pups.

c. Compute the mean and standard deviation of $X$. Interpret the mean in the context of the problem.

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.

a. Construct the probability distribution of $X$.

b. Compute the expected value $E(X)$ of $X .$ Interpret its meaning.

c. Compute the standard deviation $\sigma$ of $X$.

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. a. Construct the probability distribution of $X$. (Two entries in the table will contain $C$.) b. Compute the expected value $E(X)$ of $X$. c. Determine the value $C$ must have in order for the company to break even on all such policies (that is, to average a net gain of zero per policy on such policies). d. Determine the value $C$ must have in order for the company to average a net gain of$250 per policy on all such policies.

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.

a. Construct the probability distribution of $X$.

b. Compute the expected value $E(X)$ of $X$, and interpret its meaning in the context of the problem.

c. Compute the standard deviation of $X$.

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*}

a. Find the average time the bus takes to drive the length of its route.

b. Find the standard deviation of the length of time the bus takes to drive the length of its route.

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, $X=$ $0 .$

a. Construct the probability distribution for $X$.

b. Compute the mean $\mu$ of $X$.

c. Compute the standard deviation $\sigma$ of $X$.

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.

a. Construct the probability distribution for the number $X$ of defective units in such a sample. (A tree diagram is helpful.)

b. Find the probability that such a shipment will be accepted.

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

 Weather $x$ $P(x)$ Clear $3,000 0.61 Threatening$2,800 0.17 Light rain $1,975 0.11 Show-cancelling rain$0 0.11

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.

a. Compute the mean revenue per night if the cover is not installed.

b. Use the answer to (a) to compute the projected total revenue per 90-night season if the cover is not installed.

c. Compute the projected total revenue per season when the cover is in place. To do so assume that if the cover were in place the revenue each night of the season would be the same as the revenue on a clear night.

d. Using the answers to (b) and (c), decide whether or not the additional cost of the installation of the cover will be recovered from the increased revenue over the first ten years. Will the owner have the cover installed