Chapter 3, cont

1. Discrete Random Variables • Binomial Distribution • Geometric and Negative Binomial Distributions • Hypergeometric Distribution • Poisson Distribution Chapter 3, cont Stevan Hunter

2. Binomial Distribution • n independent Bernoulli trials • Outcome is either “success” or “failure”, no other choices • Probability “p” in each trial is the same • Binomial random variable X (= the number of trials), with parameters p and n • 0<p<1 and n = 1, 2, 3, … f(x) = (nx)px * (1-p)n-x, x = 0, 1, …, n μ = E(X) = n * p, σ2 = V(X) = n*p*(1-p) Binomial lookup table in Appendix Stevan Hunter

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5. Diagram methods of Binomial • “g” = Good, “d” = defective • {g}, {gd}, {dg}, {ggd}, {ggggggggggggggggggggd} • Tree diagram g d Stevan Hunter

6. Coin Toss • n = total number of coin tosses = 50 • x = the toss # we wish to know about • P(head) = P(tail) = 0.5 (50%) • P(X=x) = (nx)p(Head)x * p(Tail)n-x for a chosen x • P(X≤x) = F(x) = cumulative distribution function Stevan Hunter

7. Geometric Distribution • Bernoulli trials (independent, equal probability) • Random Variable X denotes the number of trials until the first Success • X is a Geometric random variable with parameter p • f(x) = (1 - p) x - 1 * p, x = 1, 2, … • μ = E(X) = 1 / p, σ2 = V(X) = (1 – p)/p2 Stevan Hunter

8. Negative Binomial Distribution • Bernoulli trials (independent, equal probability) • Random Variable X denotes the number of trials until the first r Successes • X is a Geometric random variable with parameter p • f(x) = (xr--11)(1 - p) x - r * pr, x = 1, 2, … • μ = E(X) = r / p, σ2 = V(X) = r*(1 – p)/p2 Stevan Hunter

9. Hypergeometric Distribution • Set of N objects, K successes, N-K failures • n samples are selected without replacement • (K≤N, n≤N) p = K / N • Random Variable X denotes the number of Successes in the sample • X is a Hypergeometric random variable, and • f(x) = (Kx) * (Nn--Kx) / (Nn) • x = max(0, n+K-N) to min(K,n) • μ = E(X) = n * p, where p = K / N • σ2 = V(X) = n * p * (1 - p)* (N-n) / (N – 1) Stevan Hunter

10. Poisson Distribution • Probability of > one event per subinterval = 0 • Equal probability of an event in all subintervals • The event in any subinterval is independent of all others • Random experiment within these restrictions is a Poisson Process • Random Variable X denotes the number of events in the whole interval (or the number of subintervals which have an event), with parameter 0 < λ • f(x) = (e-λ* λx ) / x!, x = 1, 2, … • μ = E(X) = λ, σ2 = V(X) = λ Stevan Hunter

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12. Poisson Distribution example • Tape noise on a continuous audio tape • E(x) = μ = λ = 20, for 200ft long tape • What is probability of having less than 15 defects in the tape? • Probability of >20? • Probability of exactly 20? • f(x) = (e-λ* λx ) / x!, x = 1, 2, … • μ = E(X) = λ, σ2 = V(X) = λ • Use Excel to make your own table Stevan Hunter

13. Poisson Distribution example Stevan Hunter