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)

For the random variables, Y and X, find α, β and γ that minimise
E[(Y−α−βX−γX^2)^2|X].
Show all derivations in your answer. You may interchangeably use
differentiation and expectation.

Let X and Y be independent random variables with means EX = 10
and EY = 5 and standard deviations σX = 2 and
σY = 1.
Find the second moment E(X + Y + 1)2

If a random variable X has a beta distribution, its probability
density function is
fX (x) = 1 xα−1(1 − x)β−1 B(α,β)
for x between 0 and 1 inclusive. The pdf is zero outside of
[0,1]. The B() in the denominator is the beta function, given by
beta(a,b) in R.
Write your own version of dbeta() using the beta pdf formula
given above. Call your function mydbeta(). Your function can be
simpler than dbeta(): use only three arguments (x, shape1,...

Let X1. ..., Xn, be a random sample from Exponential(β) with pdf
f(x) = 1/β(e^(-x/β)) I(0, ∞)(x), B > 0 where β is an unknown
parameter. Find the UMVUE of β^2.

Let α be a random variable that can take the values 1 or 2, with
probabilities 1/2 each. Let X denote the proportion of impurities
in a certain type of industrial chemicals, having a Beta(α, α + 1)
distribution. Compute E(X) and Var(X).

Let α be a random variable that can take the values 1 or 2, with
probabilities 1/2 each. Let X denote the proportion of impurities
in a certain type of industrial chemicals, having a Beta(α, α + 1)
distribution. Compute E(X) and Var(X).

Let X be a gamma random variable with parameters α > 0 and β
> 0. Find the probability density function of the random
variable Y = 3X − 1 with its support.

Let X,..., Xn be exponential with mean beta. Find
UMVUEs for beta, beta^2, beta^3. (Use the version of the
exponential distribution with PDF p(x)= 1/beta e^(-x/beta)
(x>0), and so Mx(t)=(1-beta(t))^-1.)

1. Let the angles of a triangle be α, β, and
γ, with opposite sides of length a, b,
and c, respectively. Use the Law of Cosines and the Law of
Sines to find the remaining parts of the triangle. (Round your
answers to one decimal place.)
α = 105°; b =
3; c = 10
a=
β= ____ °
γ= ____ °
2. Let the angles of a triangle be α,
β, and γ, with opposite sides of length
a, b,...

Let X have the normal distribution N(µ; σ2) and let Y = eX
(a)Find the range of Y and the pdf g(y) of Y
(b)Find the third moment of Y E[Y3]
(c) In the next four subquestions, we assume that µ = 0 and σ =
1. Sketch the graph of the pdf of Y for 0<y<=5 (use Maple to
generate the graph and copy it the best you can in the answer
box)
(d)What is the mean of Y...

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