Question

Question 1 Let X be a random variable from Poisson(X), where x > 0. That is, P(X = i) = p. = i = 0,1,2 (a) Show that P1 = P. (b) Use part (a) to write an algorithm to generate a sample from X. (c) Based on your algorithm in part (b), write an R code to generate a sample of size 1000 from Poisson (= 2). Check that the mean and the variance are of the generated sample is close to the theoretical value = 2. Report the values.

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Answer 1

# Generation from Poisson(2) distribution pois<-function L<-exp(-2) k<-0 p<-1 while (p>L) { k<-k+1 u<-runif(1) p<-pu return (k-1) X<-replicate (1000, pois ) # x is 1000 random sample from Poisson (2) mean(x) # Output : 1.944, close to theoretical value 2. var(x) # Output : 2.093 , close to theoretical value 2.

Question- X~ Poisson (2) P; = P [x=i] = -2 it lit Li = > ti (i+1) Ý => Piti = A Pi ② Algorithm Imit: L e -2 KO PI do: K -> K+ Generate PEP* u while P >L return (K-1) unu(0,1)