Question

Let n be the unknown number of customers that visit a store on the day of a sale. The number of customers that make a purchase is Y | n ~ Binomial(n, theta) where theta is the known probability of making a purchase given the customer visited the store. The prior is n ~ Poisson(5). Assuming theta is known and n is the known parameter, plot the posterior distribution of n for all combinations of Y=0,5,10 and theta=0.2,0.5 and comment on the effect of Y and theta on the posterior.

Answer 1

above is the required theory for this problem.

following is the required code in R to plot the posterior densities with varying theta and y:

following are the plots when

theta=0.2(x labels are the y values, header in the theta value)

from these above plots(when theta=0.2) with increasing y values the posterior distribution shifts to the right.

when theta=0.5(x labels are the y values, header in the theta value)

here also with increasing y the posterior distribution shifts to the right.

it is to be noted that as theta increases mode also increases for the posterior density.