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State whether each of the following random variables is discrete or continuous.
1. The number of defective tires on a car.
2. The body temperature of a hospital patient.
3. The number of pages in a book.
4. The lifetime of a lightbulb.
5. The amount of rainfall at a particular location during the next year.
6. The number of questions asked during a 1-hour lecture.
7. Airlines sometimes overbook flights. Suppose that for a plane with 100 seats, an airline takes 110 reservations. Define the variable x as the number of people who actually show up for a sold-out flight. From past experience, the probability distribution of x is given in the following table.
X 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
P(x) .05 .10 .12 .14 .24 .17 .06 .04 .03 .02 .01 .005 .005 .005 .0037 .0013
a. What is the probability that the airline can accommodate everyone who shows up for the flight?
b. What is the probability that not all passengers can be accommodated?
c. If you are trying to get a seat on such a flight and you are number 1 on the standby list, what is the probability that you will be able to take the flight? What if you are number 3?
8. A box contains five slips of paper, marked $1, $1, $1, $10, $25. The winner of a contest will select two slips of paper at random and will then get the larger of the dollar amount on the two slips. Define a random variable w by: w = amount awarded
Determine the probability distribution of w.
9. The probability distribution of x, the number of defective tires on a randomly selected automobile checked at a certain inspection station, is given in the accompanying table.
x 0 1 2 3 4
p(x) .54 .16 .06 .04 .20
a. Calculate (x, the mean value of x.
b. What is the probability that x exceeds its mean value?
10. A personal computer salesperson working on a commission receives a fixed amount for each system sold. Suppose that for a given month, the probability distribution of x = the number of systems sold is given in the accompanying table.
x 1 2 3 4 5 6 7 8
p(x) .05 .10 .12 .30 .30 .11 .01 .01
a. Find the mean value of x (the man number of systems sold.)
b. Find the variance and standard deviation of x. How would you interpret these values?
c. What is the probability that the number of systems sold is within 1 standard deviation of its mean value?
11. A local television station sells 15-sec, 30-sec, and 60-sec advertising spots. Let x denote the length of a randomly selected commercial appearing on this station, ad suppose that the probability distribution of x is given n the accompanying table.
x 15 30 60
p(x) .1 .3 .6
a. Find the average length for commercials appearing on this station.
b. If a 15-sec spot sells for $500, a 30-sec spot for $800, and a 60-sec spot for $1000, find the average amount paid for commercials appearing on this station. (Hint: Consider a new variable, y = cost, and then find the probability distribution and mean value of y.)
12. An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let x = the amount of storage space purchased by the next customer to buy a freezer. Suppose that x has the following probability distribution:
x 13.5 15.9 19.1
p(x) .2 .5 .3
a. Calculate the mean and variance of x.
b. If the price of the freezer depends on the size of the storage space, x, in the following way what is the mean value of the variable price paid by the next customer? Price = 25x - 8.5
c. What is the standard deviation of price paid?
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