Chapter Learning Objectives. Discrete Random Variables. Chapter 3: Discrete Random Variables and Probability Distributions.

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1 Chapter 3: Discrete Random Variables and Probability Distributions 3-1Discrete Random Variables ibl 3-2 Probability Distributions and Probability Mass Functions 3-33 Cumulative Distribution ib ti Functions 3-4 Mean and Variance of a Discrete Random Variable 3-5 Discrete Uniform Distribution ib i 3-6 Binomial Distribution 3-7 Geometric and Negative Binomial Distributions Geometric Distribution Negative Binomial Distribution 3-8 Hypergeometric Distribution 3-9 Poisson Distribution Chapter Learning Objectives After careful study of this chapter you should be able to do the following: 1. Determine probabilities from probability mass functions and the reverse 2. Determine probabilities from cumulative distribution functions and cumulative 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 calculate probabilities in specific applications 6. Calculate probabilities, determine means and variances for some common discrete probability distributions 1 2 Discrete Random Variables A Simple Discrete Random Variable Example Example

2 Probability Distributions and Probability Mass Functions Probability Mass Function Defined Figure 3-1 Probability distribution for bits in error. 5 6 A Probability Mass Function Example Example 3-5 A pmf Example (continued) 7 8

3 Cumulative Distribution Function Defined A Cumulative Distribution Function Example Determine the probability mass function of X from the following cumulative distribution function: 9 From either the graph or the function definition, we can see that the pmf is: f(-2)=0.2; f(0)=0.5; f(2)= Mean,Variance, and Standard Deviation of a Discrete Random Variable Example of Mean and Variance of a Discrete Random Variable Example

4 The Mean of a Function of a Discrete Random Variable We are often interested in some function of a random variable X Denoting the function of interest as h(x): The Discrete Uniform Distribution E.g., here s the pmf of a discrete uniform distribution with range 0 to 9 (incluive) E.g. as a self-study exercise, confirm that for h(x)=x 2 and for X as on the previous slide: E[h(X)] = Mean and Variance of The Discrete Uniform Distribution The Binomial Distribution E.g., for the example on the previous slide a=0 and b=9 yielding: = 4.5 and 2 =

5 Examples of Random Experiments That Might Fit The Binomial Distribution Assumptions Example pmfs for the Binomial Distribution A Numeric Binomial Distribution Example Example 3-18 A Numeric Binomial Distribution Example Example 3-18 (continued) 19 20

6 The Mean and Variance of the Binomial Distribution The Geometric Distribution Eg E.g., fore Example ample earlier we know that n=18 and p=0.1 yielding: = 1.8 and 2 = Example 3-20 A Geometric Distribution Example Example pmfs for the Geometric Distribution Also, we can compute the mean and variance using the formulae on the previous slide (with p=0.1) yielding: = 10 and 2 =

7 An Unusual Property of the Geometric Distribution Lack of Memory Property The Negative Binomial Distribution Example pmfs for the Negative Binomial Distribution The Relationship Rlti Between Bt The Geometric ti and Negative Binomial Distributions Figure Negative binomial random variable represented as a sum of geometric random variables

8 Example 3-25 A Negative Binomial Distribution Example A Negative Binomial Distribution Example Example 3-25 (continued) The Hypergeometric Distribution 31 A Hypergeometric Distribution Example 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 50, K=5 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 Binomial % 81.00% % 18.00% % 1.00% 32

9 Example pmfs for the Hypergeometric Distribution Figure Hypergeometric distributions for selected values of parameters N, K, and n. Example 3-27 Another Hypergeometric Distribution Example Another Hypergeometric Distribution Example Example 3-27 (continued) Mean and Variance of The Hypergeometric Distribution For Example 3-27 these formulae yield: = 1.33 and 2 =

10 Binomial and Hypergeometric Distributions Compared The mean for each distribution is the same (if p is interpreted as the proportion of successes in the whole batch) The variance differs only in a multiplication factor in the case of the hypergeometric: Binomial and Hypergeometric pmfs Compared 37 Figure Comparison of hypergeometric and binomial distributions. 38 The Poisson Distribution Examples of Poisson Processes In general, the Poisson random variable X is the number of events (counts) per interval : Particles of contamination per unit area Flaws per unit length of textiles Calls at a telephone exchange per unit time Power outages per unit time Atomic particles emitted from a specimen per unit time Flaws per unit length of copper wire Be careful (consistent) with units of measure when computing Poisson probabilities! bili i 39 40

11 A Poisson Distribution Example Example 3-33 Mean and Variance of The Poisson Distribution Example pmfs for this distribution: 30.0% Probability Mass Functions of the Poisson Distribution 25.0% 20.0% 15.0% 10.0% 5.0% %