chapter 5
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Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin
Chapter 5
Discrete Random Variables
5-2
Chapter Outline
5.1 Two Types of Random Variables5.2 Discrete Probability Distributions5.3 The Binomial Distribution5.4 The Poisson Distribution
(Optional)
5-3
5.1 Two Types of Random Variables
Random variable: a variable that assumes numerical values that are determined by the outcome of an experiment Discrete Continuous
Discrete random variable: Possible values can be counted or listed The number of defective units in a batch of 20 A rating on a scale of 1 to 5
5-4
Random Variables Continued
Continuous random variable: May assume any numerical value in one or more intervals The waiting time for a credit card
authorizationThe interest rate charged on a
business loan
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5.2 Discrete Probability Distributions
The probability distribution of a discrete random variable is a table, graph or formula that gives the probability associated with each possible value that the variable can assumeNotation: Denote the values of the
random variable by x and the value’s associated probability by p(x)
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Discrete Probability Distribution Properties
1
xof each valuefor 0
all
x
xp
xp
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Expected Value of a Discrete Random Variable
The mean or expected value of a discrete random variable X is:
is the value expected to occur in the long run and on average
xAll
X xpx
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Variance
The variance is the average of the squared deviations of the different values of the random variable from the expected value
The variance of a discrete random variable is:
xAll
XX xpx 22
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Standard Deviation
The standard deviation is the positive square root of the variance
The variance and standard deviation measure the spread of the values of the random variable from their expected value
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Example
Table 5.2
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Example Continued
89.
02.01.2505.01.2420.01.23
50.01.2220.01.2103.01.20
1.2
02.505.420.350.220.103.0
222
222
22
xAll xx
xAllx
xpx
xxp
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5.3 The Binomial Distribution
The binomial experiment…1. Experiment consists of n identical trials2. Each trial results in either “success” or
“failure”3. Probability of success, p, is constant from
trial to trial4. Trials are independent
If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable
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Binomial Distribution Continued
x-nxqp
x-nx
n =xp
!!
!
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Example 5.9
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x 0.05 0.1 0.15 … 0.500 0.8145 0.6561 0.5220 … 0.0625 41 0.1715 0.2916 0.3685 … 0.2500 32 0.0135 0.0486 0.0975 … 0.3750 23 0.0005 0.0036 0.0115 … 0.2500 14 0.0000 0.0001 0.0005 … 0.0625 0
0.95 0.9 0.85 … 0.50 x
Binomial Probability Table
values of p (.05 to .50)
values of p (.05 to .50)
P(x = 2) = 0.0486
p = 0.1
Table 5.7 (a)
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Several Binomial Distributions
Figure 5.6
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Mean and Variance of a Binomial Random Variable
npq
npq
np
xX
x
x
2
2
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Example
6164.38.
38.05.95.8
6.795.8
2
2
xX
x
x
npq
np
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5.4 The Poisson Distribution (Optional)
Consider the number of times an event occurs over an interval of time or space, and assume that
1. The probability of occurrence is the same for any intervals of equal length
2. The occurrence in any interval is independent of an occurrence in any non-overlapping interval
If x = the number of occurrences in a specified interval, then x is a Poisson random variable
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The Poisson Distribution Continued
!x
exp
x
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Poisson Probability Table
Table 5.9
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Poisson Probability Calculations
Table 5.10
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Mean and Variance of a Poisson Random Variable
Mean x = Variance 2
x = Standard deviation x is square
root of variance 2x
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Several Poisson Distributions
Figure 5.9
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