Question

A random spectator is to be selected from the audience of a basketball game and given the chance to shoot 10 free throws. Let Y be the number of free throws made by the selected spectator and let P be the probability with which the selected spectator makes a free throw on any attempt. Assume that P and Y follow the hierarchical model

Y |P ∼ Binomial(10, P)

P ∼ Beta(2, 2)

. (a) Run a Monte Carlo simulation to generate many realizations of Y.

Use the realizations of Y to get approximate values for

i. EY

. ii. Var Y .

(b) Find E[Y |P]

(c) Find E[Y ].

(d) Find Var[Y |P].

(e) Find Var[Y ]

. (f) Find values of α and β such that EY = 4 when P ∼ Beta(α, β).

Answer 1

(a) The following R code uses Monte Carlo simulation:

n_sim = 10000
y = replicate(n_sim, rbinom(1,10,rbeta(1,2,2)))
#E(Y):
mean(y)
#Var(Y):
sd(y)^2