Chapter 3: Discrete Random Variables and Probability Distributions

3 1 Di t R d V i bl3-1 Discrete Random Variables3-2 Probability Distributions and Probability Mass Functions3 3 C l ti Di t ib ti F ti3-3 Cumulative Distribution Functions3-4 Mean and Variance of a Discrete Random Variable3 5 Di U if Di ib i3-5 Discrete Uniform Distribution 3-6 Binomial Distribution3-7 Geometric and Negative Binomial Distributions

3-7.1 Geometric Distribution3.7.2 Negative Binomial Distribution

3-8 Hypergeometric Distribution3-9 Poisson Distribution

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Chapter Learning ObjectivesAfter careful study of this chapter you should be able to do the following:1. Determine probabilities from probability mass functions and the p p y

reverse 2. Determine probabilities from cumulative distribution functions and

cumulative distribution functions from probability mass functionscumulative distribution functions from probability mass functions, and the reverse

3. Calculate means and variances for discrete random variables 4. Understand the assumptions for some common discrete probability

distributions 5 Select an appropriate discrete probability distribution to calculate5. Select an appropriate discrete probability distribution to calculate

probabilities in specific applications 6. Calculate probabilities, determine means and variances for some

common discrete probability distributions

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Discrete Random Variables

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A Simple Discrete RandomA Simple Discrete Random Variable ExampleVariable Example

Example 3-1p

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Probability Distributions andProbability Distributions and Probability Mass FunctionsProbability Mass Functions

Figure 3-1 Probability distribution for bits in error.5

Probability Mass FunctionProbability Mass Function DefinedDefined

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A Probability Mass FunctionA Probability Mass Function ExampleExample

Example 3-5

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A pmf Example (continued)

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Cumulative Distribution FunctionCumulative Distribution Function DefinedDefined

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A Cumulative DistributionA Cumulative Distribution Function ExampleFunction Example

• Determine the probability mass function of X from the following cumulative distribution function:following cumulative distribution function:

• From either the graph or the function definition, we can see that the pmf is:that the pmf is:

f(-2)=0.2; f(0)=0.5; f(2)=0.310

Mean,Variance, and StandardMean,Variance, and Standard Deviation of a Discrete Random

Variable

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Example of Mean and VarianceExample of Mean and Variance of a Discrete Random Variableof a Discrete Random Variable

Example 3-11

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The Mean of a Function of aThe Mean of a Function of a Discrete Random Variable

• We are often interested in some function of a random variable Xrandom variable X– Denoting the function of interest as h(X):

• E.g. as a self-study exercise, confirm that for h(X)=X2 and for X as on the previous slide:( ) a d o as o t e p ev ous s de:– E[h(X)] = 158.1

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The Discrete UniformThe Discrete Uniform DistributionDistribution

•

• E.g., here’s the pmf of a discrete uniform di ib i i h 0 9 (i l i )distribution with range 0 to 9 (incluive)

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Mean and Variance of TheMean and Variance of The Discrete Uniform DistributionDiscrete Uniform Distribution

•

• E.g., for the example on the previous slide a=0 and b=9 yielding:– � = 4.5 and �2 = 8.25

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The Binomial Distribution

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l f d i h i h iExamples of Random Experiments That Might Fit The Binomial Distribution Assumptions

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Example pmfs for the BinomialExample pmfs for the Binomial DistributionDistribution

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A Numeric Binomial DistributionA Numeric Binomial Distribution ExampleExample

Example 3-18

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A Numeric Binomial DistributionA Numeric Binomial Distribution ExampleExample

Example 3-18 (continued)

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The Mean and Variance of theThe Mean and Variance of the Binomial DistributionBinomial Distribution

• E g for E ample 3 18 earlier e kno that• E.g., for Example 3-18 earlier we know that n=18 and p=0.1 yielding:– � = 1.8 and �2 = 1.62

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The Geometric Distribution

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A Geometric DistributionA Geometric Distribution ExampleExample

Example 3-20

• Also, we can compute the mean and variance using the formulae on the previous slide (with p=0 1) yielding:

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formulae on the previous slide (with p=0.1) yielding:– � = 10 and �2 = 90

Example pmfs for the GeometricExample pmfs for the Geometric Distribution

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An Unusual Property of theAn Unusual Property of the Geometric Distribution

Lack of Memory Property

Geometric Distribution

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The Negative BinomialThe Negative Binomial DistributionDistribution

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Example pmfs for the NegativeExample pmfs for the Negative Binomial DistributionBinomial Distribution

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Th R l ti hi B t Th G t i dThe Relationship Between The Geometric and Negative Binomial Distributions

Figure 3-11. Negative binomial random variable represented as a sum of geometric random variablesrepresented as a sum of geometric random variables.

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A Negative BinomialA Negative Binomial Distribution ExampleDistribution Example

Example 3-25

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A Negative BinomialA Negative Binomial Distribution ExampleDistribution Example

Example 3-25 (continued)

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The Hypergeometric Distribution

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A Hypergeometric DistributionA Hypergeometric Distribution ExampleExample

• Modifying Example 3-18, suppose we have a batch of 50 samples of water, 5 of which are contaminated

• If we draw a random sample of size 2, without replacement, what’s the distribution of the number of contaminated samples?

• This is a hypergeometric random variable with N=50 K=5 and n=2This is a hypergeometric random variable with N 50, K 5, and n 2(not a binomial random variable with p=0.1 and n=2)

• Using equations 3-13 and 3-7 to determine pmf values yields:

f(x)x Hypergeometric Binomialyp g

0 80.82% 81.00%1 18 37% 18 00%

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1 18.37% 18.00%2 0.82% 1.00%

Example pmfs for theExample pmfs for the Hypergeometric Distribution

Figure 3-12. Hypergeometric

Hypergeometric Distribution

Hypergeometric distributions for selected values of parameters N, K, and n.

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Another HypergeometricAnother Hypergeometric Distribution ExampleDistribution Example

Example 3-27

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Another HypergeometricAnother Hypergeometric Distribution Example

Example 3-27 (continued)Distribution Example

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Mean and Variance of TheMean and Variance of The Hypergeometric DistributionHypergeometric Distribution

• For Example 3-27 these formulae yield:– � = 1.33 and �2 = 0.88

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� 1.33 and � 0.88

Binomial and HypergeometricBinomial and Hypergeometric Distributions ComparedDistributions Compared

• The mean for each distribution is the same (if p is i d h i f “ ” i hinterpreted as the proportion of “successes” in the whole batch)

• The variance differs only in a multiplication factor in the case of the hypergeometric:

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Binomial and HypergeometricBinomial and Hypergeometric pmfs Comparedpmfs Compared

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Figure 3-13. Comparison of hypergeometric and binomial distributions.

The Poisson Distribution

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Examples of Poisson Processes• In general, the Poisson random variable X is the

number of events (counts) per “interval”:number of events (counts) per interval :– Particles of contamination per unit area

Flaws per unit length of textiles– Flaws per unit length of textiles– Calls at a telephone exchange per unit time

Power outages per unit time– Power outages per unit time– Atomic particles emitted from a specimen per unit time

Fl it l th f i– Flaws per unit length of copper wire

• Be careful (consistent) with units of measure when i P i b bili i !computing Poisson probabilities!

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A Poisson Distribution ExampleA Poisson Distribution ExampleExample 3-33

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Mean and Variance of The Poisson Distribution

• Example pmfs for this distribution:

30.0%

Probability Mass Functions of the Poisson Distribution

20.0%

25.0%

10.0%

15.0% ������

420.0%

5.0%

0 2 4 6 8 10 12 14 16 18 20