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Engineering Statistics - IE 261

Chapter 3Discrete Random Variables andProbability Distributions

URL: http://home.npru.ac.th/piya/ClassesTU.htmlhttp://home.npru.ac.th/piya/webscilab

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3-1 Discrete Random Variables

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3-1 Discrete Random Variables

Example 3-1

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3-2 Probability Distributions and Probability Mass Functions

Figure 3-1 Probability distribution for bits in error.

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3-2 Probability Distributions and Probability Mass FunctionsDefinition

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Example 3-5

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Example 3-5 (continued)

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3-3 Cumulative Distribution Functions

Definition

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Example 3-8

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Example 3-8

Figure 3-4 Cumulative distribution function for Example 3-8.

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3-4 Mean and Variance of a Discrete Random VariableDefinition

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3-4 Mean and Variance of a Discrete Random Variable

Figure 3-5 A probability distribution can be viewed as a loading with the mean equal to the balance point. Parts (a) and (b) illustrate equal means, but Part (a) illustrates a larger variance.

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Proof of Variance:

22 V X E X

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3-4 Mean and Variance of a Discrete Random Variable

Figure 3-6 The probability distribution illustrated in Parts (a) and (b) differ even though they have equal means and equal variances.

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Example 3-9There is a chance that a bit transmitted through a digital transmissionchannel is received in error. Let X equal the number of bits in error inthe next four bits transmitted. The possible values for X are {0, 1, 2, 3, 4}Suppose:

P(X = 0) = 0.6561 P(X = 1) = 0.2916 P(X = 2) = 0.0486

P(X = 3) = 0.0036 P(X = 4) = 0.0001

Find the mean and the variance of X

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Example 3-9 (Solution)

( )E X

SCILAB-->x = [0 1 2 3 4];-->fx = [0.6561 0.2916 0.0486 0.0036 0.0001];-->MeanX = sum(x.*fx) MeanX = 0.4 -->VarX = sum((x.^2).*fx) - MeanX^2 VarX = 0.36

k kk

x f x

2 2 2k k

k

x f x

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Example 3-9 (Solution)

2 ( )V X

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Example 3-11

-->x = [10:15]; fx = [0.08 0.15 0.3 0.2 0.2 0.07];-->MeanX = sum(x.*fx) MeanX = 12.5-->VarX = sum((x.^2).*fx) - MeanX^2 VarX = 1.85