Question

Let X ∼ Beta(α, β).

(a) Show that EX 2 = (α + 1)α (α + β + 1)(α + β) .

(b) Use the fact that EX = α/(α + β) and your answer to the previous part to show that Var X = αβ (α + β) 2 (α + β + 1).

(c) Suppose X is the proportion of free-throws made over the lifetime of a randomly sampled kid, and assume that X ∼ Beta(2, 8)

. i. Draw a picture of the pdf of X (you may use software if you wish).

ii. Compute the probability that the randomly selected kid has made fewer than 0.10 of all free throws. Hint: Use the R function pbeta(). Access documention with ?pbeta.

iii. If you were to sample 100 kids and compute the average of their lifetime free-throw success proportions, to what number would you expect this average to be close?

(d) Let α = 1 and β = 1. For some constants a and b with a < b, let Y = (b − a)X + a.

Find {A useful fact: x 3 − y 3 = (x − y)(x 2 + xy + y 2 ) for any x, y.]

i. the support of the new random variable Y .

ii. EY .

iii. Var Y .

(e) Give the pdf of X under α = 1 and β = 1.

(f) Give the cdf of X under α = 1 and β = 1.

(g) What is the relationship between the beta distribution and the uniform distribution?

Answer 1

(i) R code:

x=seq(0,1,0.0101)
plot(x,dbeta(x,2,8),type="l",xlab="x",ylab="f(x)",main="PDF of Beta(2,8)")

(ii)

R code: pbeta(0.10,2,8)

Output: 0.225159

(iii) Expected average=100*(2/10)=20.

(d) Y=(b-a)X+a; since 0<X<1a<Y<b.

(ii) E(Y)=(b-a)E(X)+a=(b-a)*(1/2)+a=(a+b)/2

(iii)