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Chapter 5 Discrete Probability Distributions

Learning Objectives

1. Understand the concepts of a random variable and a probability distribution.

2. Be able to distinguish between discrete and continuous random variables.

3. Be able to compute and interpret the expected value, variance, and standard deviation for a discrete random variable.

4. Be able to compute and work with probabilities involving a binomial probability distribution.

5. Be able to compute and work with probabilities involving a Poisson probability distribution.

A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals.

Discrete Probability Distributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Distribution n Poisson Distribution

[pic] A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals.

Example: JSL Appliances n Discrete random variable with a finite number of values n Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4)

n Discrete random variable with an infinite sequence of values n Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2, . . . n We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

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Some Examples:

Example 34:Forty percent of Business travelers carry either cell phone or a laptop. For a sample of 15 business travelers, make the following calculations. a) Compute the probability that three of the travelers carry a cell phone or a laptop. b) Compute the probability that 12 of the travelers carry neither a cell phone nor a laptop c) Compute the probability that at least three of the travelers carry a cell phone or a laptop

Solution a) P(3) = .0634 (from tables)

(b) The answer here is the same as part (a). The probability of 12 failures with p = .60 is the same as the probability of 3 successes with p = .40.

(c). p (3) + p (4) + · · · + p (15) =1 - p (0) - p (1) - p (2) = 1 - .0005 - .0047 - .0219 = .9729

Problem set Chapter 5

1. The manager of a computer repair center in Trenton, New Jersey. has read an advertisement from a local competitor that guarantee. His past service records are used to determine the following probability distribution.
|Number of days |Probability |
|1 |0.15 |
|2 |0.25 |
|3 |0.30 |
|4 |0.18 |
|5 |0.12 |

a. Calculate the mean number of days his customers wait for a computer repair.

b. Also calculate the variance and standard deviation.

c. Based on the calculations in parts a and b, what calculations in parts a and b, what conclusion should the manager reach regarding his company’s repair times?

2. For a binomial distribution with a sample size equal to 10 and a probability of a success equal to 0.30, what is the probability that the sample will contain exactly three successes? Use the calculator to determine the probability.

3. A bank manager has recently examined the credit card account balances for the customers of her bank and found that 20% have an outstanding balance that is at the credit card limit. Suppose the manager randomly selects 15 customers and finds 4 that have balances at the limit. Assume that the properties of the binomial distribution apply. a. What is the probability of finding 4 customers in a sample of 15 who have "maxed out" their credit cards? b. What is the probability that 4 or fewer customers in the sample will have balances at the limit of the credit card?

4. Use the binomial formula to calculate the following probabilities for an experiment in which n = 5 and p = 0.4: a. the probability that x is at most 1 b. the probability that x is at least 4 c. the probability that x is less than 1

5. Studies indicate that 40% of all home buyers will do some remodeling to their home within the first five years of home ownership. If this is true, use the binomial distribution to determine the probability that in a random sample of 20 homeowners, two or fewer will remodel their homes.

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