MATH 1LT3 Chapter : binomial.pdf

• School: McMaster University
• Department: Mathematics
• Course: MATH 1LT3
• Professor: Erin Clements

Document Summary

The probability function for a binomial random variable is (cid:0) (cid:1) n x b(x; n; p) = px(1 p)n x. This is the probability of having x successes in a series of n independent trials when the probability of success in any one of the trials is p. If x is a random variable with this probability distribution, nx nx nx x=0 x=0 x=1. E(x) = (cid:0) (cid:1) px(1 p)n x x n x x n! x!(n x)! px(1 p)n x (x 1)!(n x)! px(1 p)n x n! since the x = 0 term vanishes. Let y = x 1 and m = n 1. Subbing x = y +1 and n = m+1 into the last sum (and using the fact that the limits x = 1 and x = n correspond to y = 0 and y = m, respectively) mx y=0. E(x) = (m+1)! mx y!(m y)! py+1(1 p)m mx y=0 y!(m y)! py(1 p)m m!