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Classnotes_5 - 2 Discrete Random Variables 3 A Simple...

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Unformatted text preview: 2 Discrete Random Variables 3 A Simple Discrete Random Variable Example Example 3-1 p 4 Probability Distributions and Probability Mass Functions Figure 3-1 Probability distribution for bits in error. 5 Probability Mass Function Defined 6 A Probability Mass Function Example Example 3-5 7 A pmf Example (continued) 8 Cumulative Distribution Function Defined 9 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   10 Mean,Variance, and Standard Deviation of a Discrete Random Variable 11 Example of Mean and Variance of a Discrete Random Variable Example 3-11 12 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. 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 The Discrete Uniform Distribution • • 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) 14 Mean and Variance of The Discrete Uniform Distribution • • E.g., for the example on the previous slide D  and E  yielding: – ȝ = 4.5 and ı2 = 8.25 15 The Binomial Distribution 16 Examples of Random Experiments That Might Fit l f d i h i h i The Binomial Distribution Assumptions 17 Example pmfs for the Binomial Distribution 18 A Numeric Binomial Distribution Example Example 3-18 19 A Numeric Binomial Distribution Example Example 3-18 (continued) 20 The Mean and Variance of the Binomial 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 The Geometric Distribution 22 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 Example pmfs for the Geometric Distribution 24 An Unusual Property of the Geometric Distribution Lack of Memory Property 25 The Negative Binomial Distribution 26 Example pmfs for the Negative Binomial Distribution 27 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. 28 A Negative Binomial Distribution Example Example 3-25 29 A Negative Binomial Distribution Example Example 3-25 (continued) 30 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, 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) 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 Figure 3-12. Hypergeometric distributions for selected values of parameters N, K, and n. 33 Another Hypergeometric Distribution Example Example 3-27 34 Another Hypergeometric Distribution Example Example 3-27 (continued) 35 Mean and Variance of The Hypergeometric Distribution • For Example 3-27 these formulae yield: – ȝ = 1.33 and ı2 = 0.88 36 Binomial and Hypergeometric Distributions 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 Binomial and Hypergeometric pmfs Compared Figure 3-13. Comparison of hypergeometric and binomial distributions. 38 The Poisson Distribution 39 A Poisson Distribution Example Example 3-33 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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This note was uploaded on 05/31/2010 for the course ENGG 319 taught by Professor Nanayakkara during the Fall '08 term at University of Calgary.

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