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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 Bernoulli Trials • Examples: 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”) Administer a test drug to a group of patients with a specific condition. P(“success”) = ___________ • 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. 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. Binomial Distribution • The probability distribution is called the binomial distribution. n x nx b(x; n, p) = p q x , x = 0, 1, 2, …, n where p = _________________ q = _________________ • For our example, b(x; n, p) = _________________ 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? Your turn … • What is the probability that more than 2 of the 4 bits are in error? 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 = ___________ σ2 = __________ , σ = __________ 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? What is the expected variance of the 25 parts? Helpful Hints … • Sometimes it helps to draw a picture. Suppose we inspect the next 5 parts … P(at least 3) P(2 ≤ X ≤ 4) P(less than 4) • Appendix Table A.1 (pp. 661-666) lists Binomial Probability Sums, ∑rx=0b(x; n, p) Your turn … • Use Table A.1 to determine 1. b(x; 15, 0.4) , P(X ≤ 8) = ______________ 2. b(x; 15, 0.4) , P(X < 8) = ______________ 3. b(x; 12, 0.2) , P(2 ≤ X ≤ 5) = ___________ 4. b(x; 4, 0.1) , P(X > 2) = ______________ Multinomial Experiments • What if there are more than 2 possible outcomes? (e.g., acceptable, scrap, rework) • That is, suppose we have: n independent trials k outcomes that are mutually exclusive (e.g., ♠, ♣, ♥, ♦) exhaustive (i.e., ∑all kpi = 1) • Then n x1 x2 p1 p2 ... pkxk f(x1, x2, …, xk; p1, p2, …, pk, n) = x1, x2 ,..., xk Example • Look at problem 22, pg. 126 x1 = _______ p1 = _______ x2 = _______ p2 = _______ x3 = _______ p3 = _______ n = _____ f( __, __, __; ___, ___, ___, __) =_________________ = __________________________________