Known Probability Distributions

1. Known Probability Distributions • Engineers frequently work with data that can be modeled as one of several known probability distributions. • Being able to model the data allows us to: • model real systems • design • predict results • Key discrete probability distributions include: • binomial • negative binomial • hypergeometric • Poisson EGR 252.001 Spring 2011

2. Discrete Uniform Distribution • Simplest of all discrete distributions • All possible values of the random variable have the same probability, i.e., f(x; k) = 1/ k, x = x1 , x2 , x3 , … , xk • Expectations of the discrete uniform distribution EGR 252.001 Spring 2011

3. Binomial & Multinomial Distributions • Bernoulli Trials • Inspect tires coming off the production line. Classify each as defective or not defective. Define “success” as defective. If historical data shows that 95% of all tires are defect-free, then P(“success”) = 0.05. • Signals picked up at a communications site are either incoming speech signals or “noise.” Define “success” as the presence of speech. P(“success”) = P(“speech”) • Bernoulli Process • n repeated trials • the outcome may be classified as “success” or “failure” • the probability of success (p) is constant from trial to trial • repeated trials are independent EGR 252.001 Spring 2011

4. Binomial Distribution • Example: Historical data indicates that 10% of all bits transmitted through a digital transmission channel are received in error. Let X = the number of bits in error in the next 4 bits transmitted. Assume that the transmission trials are independent. What is the probability that • Exactly 2 of the bits are in error? • At most 2 of the 4 bits are in error? • More than 2 of the 4 bits are in error? • The number of successes, X, in n Bernoulli trials is called a binomial random variable. EGR 252.001 Spring 2011

5. Binomial Distribution • The probability distribution is called the binomial distribution. • b(x; n, p) = , x = 0, 1, 2, …, n where p = probability of success q = probability of failure = 1-p For our example, • b(x; n, p) = EGR 252.001 Spring 2011

6. For Our Example … • What is the probability that exactly 2 of the bits are in error? • At most 2 of the 4 bits are in error? • More than 2 of the 4 bits are in error? EGR 252.001 Spring 2011

7. Expectations of the Binomial Distribution • The mean and variance of the binomial distribution are given by μ =np σ2 = npq • Suppose, in our example, we check the next 20 bits. What are the expected number of bits in error? What is the standard deviation? μ = 20 (0.1) = 2 σ2 = 20 (0.1) (0.9) = 1.8σ = 1.34 EGR 252.001 Spring 2011

8. Another example • A worn machine tool produces 1% defective parts. If we assume that parts produced are independent, what is the mean number of defective parts that would be expected if we inspect 25 parts? • μ = 25 (0.01) = 0.25 • What is the expected variance of the 25 parts? • σ2 = 25 (0.01) (0.99) = 0.2475 • Note that 0.2475 does not equal 0.25. EGR 252.001 Spring 2011

9. Helpful Hints … • Suppose we inspect the next 5 parts …b(x ; 5, 0.01) • Sometimes it helps to draw a picture. P(at least 3)  ________________ 0 1 2 3 4 5 P(2 ≤ X ≤ 4)  ________________ 0 1 2 3 4 5 P(less than 4)  ________________ 0 1 2 3 4 5 • Appendix Table A.1 (pp. 742-747) lists Binomial Probability Sums, ∑rx=0 b(x; n, p) EGR 252.001 Spring 2011