Distribution Function of Random Variables Definition A cumulative

Distribution Function of Random Variables • Definition A cumulative distribution function (cdf) of a random variable X is a mapping F: R [0, 1] defined by • If X is a discrete random variable with pmf where for x = 0, 1, 2, … then is the greatest integer ≤ v. • Example: week 3 1

Properties of Distribution Function • F is monotone, non decreasing i. e. F(x) ≤ F(y) if x ≤ y. • As x - ∞ , F(x) 0 • As x ∞ , F(x) 1 • F(x) is continuous from the right • For a < b Why? week 3 2

Exercises 1. A box contain 20 notes numbered 20 to 39. We randomly pick one note and record its number. What is the probability that the number we got is greater then 32? 2. 30% of U of T students wear glasses. We select a random sample of size 10 students. a) What is the probability that exactly 4 of them wear glasses? b) What is the probability that more then 3 wear glasses? 3. We roll a die until we obtained an even outcome. a) What is the probability that we will roll the die exactly 5 times? b) What is the probability that we roll the die more then 7 times ? c) What is the probability that we roll the die more then 7 times if we know that we need more then 2 rolls? week 3 3

4. We roll a die until we get 6 even outcomes. a) What is the probability that we need exactly 10 rolls? b) What is the probability that we need less 10 rolls? 5. The number of cars that cross Spadina and Bloor intersection is a Poisson random variable with λ = 15 cars per minute. a) What is the probability that in a given minute exactly 15 cars will cross the intersection? b) What is the probability that in a given minute more then 15 cars will cross the intersection? c) What is the probability that during half an hour there where exactly 2 minutes in which 15 cars crossed the intersection? week 3 4

Relation between Binomial and Poisson Distributions • Binomial distribution Model for number of success in n trails where P(success in any one trail) = p. • Poisson distribution is used to model rare occurrences that occur on average at rate λ per time interval. Can think of “rare” occurrence in terms of p 0 and n ∞. Take these limits so that λ = np. • So we have that week 3 5