Classnotes_3 - 2 A Simple Discrete Random Variable Example...

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Unformatted text preview: 2 A Simple Discrete Random Variable Example Discrete Random Variables Example 3-1 p 3 4 Probability Distributions and Probability Mass Functions Probability Mass Function Defined Figure 3-1 Probability distribution for bits in error. 5 A Probability Mass Function Example 6 A pmf Example (continued) Example 3-5 7 8 Cumulative Distribution Function Defined A Cumulative Distribution Function Example • Determine the probability mass function of X from the following cumulative distribution function: • From either the graph or the function definition, we can see that the pmf is: I   I   I   9 Mean,Variance, and Standard Deviation of a Discrete Random Variable 10 Example of Mean and Variance of a Discrete Random Variable Example 3-11 11 12 The Discrete Uniform Distribution 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): • E.g., here’s the pmf of a discrete uniform distribution i h di ib i with range 0 to 9 (i l i ) (incluive) • E.g. as a self-study exercise, confirm that for ( ) a d o o t e p ev ous s de: h(X)=X2 and for X as on the previous slide: – E[h(X)] = 158.1 13 Mean and Variance of The Discrete Uniform Distribution 14 The Binomial Distribution • • E.g., for the example on the previous slide D  and E  yielding: – ȝ = 4.5 and ı2 = 8.25 15 16 Example pmfs for the Binomial Distribution Examples of Random Experiments That Might Fit l f d i h i h i The Binomial Distribution Assumptions 17 A Numeric Binomial Distribution Example 18 A Numeric Binomial Distribution Example Example 3-18 (continued) Example 3-18 19 20 The Mean and Variance of the Binomial Distribution The Geometric Distribution • E g for Example 3-18 earlier we know that E.g., E ample 3 18 e kno Q  and S  yielding: – ȝ = 1.8 and ı2 = 1.62 21 22 Example pmfs for the Geometric Distribution A Geometric Distribution Example Example 3-20 • Also, we can compute the mean and variance using the formulae on the previous slide (with S  ) yielding: ) – ȝ = 10 and ı2 = 90 23 24 An Unusual Property of the Geometric Distribution The Negative Binomial Distribution Lack of Memory Property 25 Example pmfs for the Negative Binomial Distribution 26 The R l ti hi B t Th Relationship Between The Geometric and Th G ti d Negative Binomial Distributions Figure 3-11. Negative binomial random variable represented as a sum of geometric random variables variables. 27 28 A Negative Binomial Distribution Example A Negative Binomial Distribution Example Example 3-25 Example 3-25 (continued) 29 30 A Hypergeometric Distribution Example The Hypergeometric Distribution • 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, from these 50, what’s the distribution of the number of contaminated samples? N 50, K 5, n 2 • This 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) 31 x 0 1 2 Hypergeometric yp g 80.82% 18.37% 18 37% 0.82% Binomial 81.00% 18.00% 18 00% 1.00% 32 Example pmfs for the Hypergeometric Distribution Another Hypergeometric Distribution Example Figure 3-12. Hypergeometric distributions for selected values of parameters N, K, and n. Example 3-27 33 Another Hypergeometric Distribution Example 34 Mean and Variance of The Hypergeometric Distribution Example 3-27 (continued) • For Example 3-27 these formulae yield: – ȝ = 1.33 and ı2 = 0.88 35 36 Binomial and Hypergeometric Distributions Compared Binomial and Hypergeometric pmfs Compared • The mean for each distribution is the same (if p is interpreted as the proportion of “successes” in the i d h i f“ ”i h whole batch) • The variance differs only in a multiplication factor in the case of the hypergeometric: 37 Figure 3-13. Comparison of hypergeometric and binomial distributions. 38 A Poisson Distribution Example The Poisson Distribution Example 3-33 39 40 Mean and Variance of The Poisson Distribution • Example pmfs for this distribution: Probability Mass Functions of the Poisson Distribution 30.0% 25.0% 20.0% O  O  15.0% 10.0% 5.0% 0.0% 0 2 4 6 8 10 12 14 16 18 20 41 42 ...
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