The Poisson random variable is a discrete random variable that models how many t

Question

The Poisson random variable is a discrete random variable that models how many taxis come by in a fixed time, or how many charged particles are detected in a fixed time, or how hard drive crashes in a fixed time. The probability mass function of the Poisson random variable with mean (and variance) lambda is f(x) = {lambda^x e^-lambda/x!, x = 0, 1... 0, otherwise. Write an R program to generate i.i.d. Poisson random variables with mean one, X_1,...,X_n from uniform random variables, Z_1,...,Z_n using the inverse distribution transformation method. Use your program to print out Z_1,...,Z_n and X_1,...,X_n for n = 10.

Explanation / Answer

The function takes as input some value x and tells you what is the probability of obtaining Xx

So, FX(x)=Pr(Xx)=p and thus F-1(p) = x

The idea is very simple: it is easy to sample values uniformly from U(0,1), so if you want to sample from some FX, just take values uU(0,1) and pass u through F1X to obtain x's, F1X(u) = x

Algorithm for generating a Poisson rv X with mean

i Set X = 0, P = 1
ii Generate U unif (0,1), set P = U P
iii If P < e , then stop. Otherwise if P e , then set X = X + 1 and go back to ii.

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