STAT 100A Lecture Notes - Lecture 12: Poisson Distribution, Random Variable
Document Summary
A random variable x that takes on whole number values is said to be a poisson random variable with parameter if for some > 0, P(x = i) = e i i! where i w. If n , the probability distribution of x is a poisson distribution. P(x = i) = (cid:2)n i(cid:3)pi(1 p)n i n! (n i)!i! pi(1 p)n i. P(x = i) = n! (n i)!i! (cid:2) n(cid:3)i(cid:2)1 n(cid:3)n i (n)(n 1) . (n i + 1) ni. I i! n(cid:5)2 (cid:4)1 n(cid:5)i (cid:4)1 . E[x] = i (cid:6)i=0 (cid:6)i=1 e i i! e i 1 (i 1)! As represents the total probability of x, e j j! (cid:7)j=0 (cid:6)j=0 e j j! E(cid:8)x 2(cid:9) = i2 e i i! i e i 1 (i 1)! (cid:6)i=0 (cid:6)i=1. Therefore, e j j! e j (j + 1) (cid:6)j=0.
