代写Statistics 100B、代做Python/Java编程、代写Statistics留学生

日期：2019-01-18 10:15

University of California, Los Angeles

Department of Statistics

Statistics 100B Instructor: Nicolas Christou

Homework 2

EXERCISE 1

Let X ～ N(μ, σ).

a. Use the properties of moment generating functions to show that aX + b ～ N(aμ + b, aσ).

b. Use the cdf method to show that aX + b ～ N(aμ + b, aσ).

EXERCISE 2

Answer the following questions:

a. Let ln(X) ～ N(μ, σ). Find EX and var(X).

b. Let X1, X2, . . . , Xn be independent random variables having respectively the normal distributions

N(μi

, σi), i = 1, . . . , n. Consider the random variable Y =

Pn

i=1 kiXi

. Use moment generating functions

to find the distribution of Y .

c. Let X1, X2, . . . , Xn be i.i.d. random variables with Xi ～ Γ(α, β). Use the properties of moment generating

functions to find the distribution of T = X1 + X2 + . . . Xn and Xˉ =

X1+X2+...Xn

n

.

EXERCISE 3

Let X ～ N(μ, σ). Stein’s lemma states that if g is a differentiable function satisfying Eg0

(X) < ∞ then

E [g(X)(X ? μ)] = σ

2Eg0

(X). Use Stein’s lemma to show that EX4 = μ

4 + 6μ

2σ

2 + 3σ

4

. Hint: Write EX4

as EX3

(X  μ + μ).

EXERCISE 4

Let X1, . . . , Xn i.i.d. random variables with Xi ～ N(μ, σ). Express the vector

in the form

AX and find its mean and variance covariance matrix. Show some typical elements of the variance covariance

matrix.

EXERCISE 5

Answer the following questions:

a. Suppose X has a uniform distribution on (0, 1). Find the mean and variance covariance matrix of the

random vector 

b. Suppose X1 and X2 are independent with Γ(α, 1) and Γ(α +

1

2

, 1) distributions. Let Y = 2√

X1X2.

Find EY and var(Y ).

EXERCISE 6

Answer the following questions.

a. Let X = (X1, . . . , Xn)

0 be a random vector with joint moment generating function MX(t). In class

we discuss this theorem: Let Mi(t) = MX(t)

ti

, Mii(t) =

2MX(t)

ti

2 , and Mij (t) =

2MX(t)

ti tj

. Then,

EXi = Mi(0), EX2

i = Mii(0), and EXiXj = Mij (0). Prove this theorem when n = 2.

b. Suppose U ～ Γ(α, β), with α > 0, β > 0 and let Y = e

U . Find the probability density function of Y .

Find EY and var(Y )