JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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 = x 1, x 2, x 3, …, x k Expectations of the discrete uniform distribution
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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”) = 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
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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.
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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) =
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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?
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 7 Expectations of the Binomial Distribution The mean and variance of the binomial distribution are given by μ = np σ 2 = 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 σ 2 = 20 (0.1) (0.9) = 1.8 σ = 1.34
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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? σ 2 = 25 (0.01) (0.99) = Note that does not equal 0.25.
JMB Chapter 5 Part 1 EGR Spring 2011 Slide 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) ________________ P(2 ≤ X ≤ 4) ________________ P(less than 4) ________________ Appendix Table A.1 (pp ) lists Binomial Probability Sums, ∑ r x=0 b(x; n, p)