9. The short answer is that the Poisson approximation is faster and easier to compute and reason about, and among other things tells you approximately how big the exact answer is. Here's a simple example: suppose you're trying to get something to happen in a video game that is rare; maybe it happens 1% of the time you do something.. The approximation theorems. Let Xi indicate success on the ith trial, so that P(Xi = 1) -pi and P(Xi = 0) = 1 - pi. Our proofs will be based. on the device of introducing random variables Yi that have the Poisson dis-. tribution with E( Yi) = pi, and are such that P(Xi = Yi) is as large as pos-. sible.
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Based on the connection between the Binomial and Poisson distributions it intuitively makes sense that we should also be able to approximate the Poisson with a Normal distribution. For approximation to the binomial we need np 10 and nq 10. What is a reasonable requirement for ? Statistics 104 (Colin Rundel) Lecture 7 February 6, 2012 11 / 26. Poisson approximations 9.1Overview The Bin(n;p) can be thought of as the distribution of a sum of independent indicator random variables X 1 + + X n, with fX i= 1gdenoting a head on the ith toss of a coin that lands heads with probability p. Each X i has a Ber(p) distribution. The normal approximation to the Binomial works
