Poisson/Binomial distribution?

Posted on 2016-03-03 18:43:00 in Math by HAYDEN

Poisson distribution with parameter lambda approximates the Binomial distribution with parameters n and p such that lambda = np if n is very big, and p is very small. Some textbooks suggest that when np less than 5, the Binomial distribution is approximated well enough. Assume you have a binomial random variable x with n = 20, and the probability of success of p = 0.25. You approximate it with y which is distributed with a Poisson distribution with a parameter of l = 5 = np. Fill up the table: 
Value k 0 2 4 6 8 10 12 14 16 18 20 >20 
P(x = k) 
P(y = k) 
What are the theoretical means of x and y? What are their theoretical variances? How would you argue that y approximates x well?

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