1. JMB Chapter 5 Part 1 EGR 252.001 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

2. JMB Chapter 5 Part 1 EGR 252.001 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

3. JMB Chapter 5 Part 1 EGR 252.001 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”) = 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

4. JMB Chapter 5 Part 1 EGR 252.001 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.

5. JMB Chapter 5 Part 1 EGR 252.001 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) =

6. JMB Chapter 5 Part 1 EGR 252.001 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?

7. JMB Chapter 5 Part 1 EGR 252.001 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

8. JMB Chapter 5 Part 1 EGR 252.001 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) = 0.2475 Note that 0.2475 does not equal 0.25.

9. JMB Chapter 5 Part 1 EGR 252.001 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)  ________________ 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. 726-731) lists Binomial Probability Sums, ∑ r x=0 b(x; n, p)