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# Lec07 - Common Discrete Distributions IOE/Stat 265 Fall...

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1 Ch. 3.4-3.5 IOE/Stat 265, Fall 2009 Lecture #7: Lecture #7: Binomial Distribution (and Relatives) Binomial Distribution (and Relatives) 2 Common Discrete Distributions Discrete Uniform 3.4 Binomial Special Case: Bernoulli Geometric Negative Geometric 3.5 Hypergeometric & Negative Binomial 3.6 Poisson 3 Discrete Uniform Distribution ± A random variable X has a discrete uniform distribution if p(x) = 1/n for x 1 , x 2 , …, x n () + = + μ= = +− σ= −μ = K 2 22 1 a,a 1, ,b p(x) n 0 otherwise ba E(x) 2 ba1 1 E(x ) 12 024681 0 X 0 0.05 0.1 0.15 0.2 Discrete Uniform Probability Mass Function a=1 b=10 4 Example 1: Search & Rescue ± It’s 40 miles to DTW airport. Your car has broken down on the way to catch a plane. You call a tow truck for assistance and describe your location based on a nearby mile marker. ± What is the probability that you are less than 10 miles from the airport? ± What is the standard deviation of distance?

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7 3-4 Bernoulli Distribution ± Most fundamental distribution: The number of successes in one Bernoulli trial. We assume the probability of success, p, is known and that the trial can result in only one of two mutually exclusive outcomes – success or failure. ± Bernoulli trial? A random experiment with only two possible outcomes, "success" and "failure" with probabilities p and (1-p). ± p(x) = p x (1 – p) 1-x x = 0, 1 0 p 1 ± μ = E(x) = p σ 2 = E(x- μ) 2 = p (1–p) 8 p = 0.1 p = 0.2 p = 0.5 0 0.2 0.4 0.6 0.8 1 X 0 0.2 0.4 0.6 0.8 1 Bernoulli Probability Mass Function 9 Example 2: Defective Parts ² Suppose 0.1% of production parts are defective. What is the probability that the first item inspected will be nondefective? 11 3-4 Binomial Distribution ² A binomial random variable arises with “n” repeated trials of a Bernoulli experiment. ² Characteristics: ² n trials, fixed in advance ² Identical trials, with only 2 possible outcomes (Bernoulli trial) ² Independent trials ² Probability of success (p) is constant
12 Examples ± Flipping a coin 10 times ±

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Lec07 - Common Discrete Distributions IOE/Stat 265 Fall...

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