 Statistics统计学作业代写

## Assignment 1

Statistics统计学作业代写 Suppose that ~ (0,1). Let Y be independent of X with P(Y = 1) = P(Y = -1) = 1/2. Define the random variable Z be setting Z = XY.

Each question is worth 10 marks for a total of 40 marks. Question 5 is a bonus question. 1.

Let X and Y be independent random variables with ~ (1,3) and Y~ (0,1).

### 1.    Statistics统计学作业代写

a)Determine ,( , ), the joint distribution function of (X, Y)’.

b)Show directly (by computing the  indicated partialderivative) that: ,  (  ,   ) = ( ) ( )

c)Is this surprising? Why or whynot?

d)If Z~ (4).is independent of X and Y, determine the joint density of (X, Y, Z)’.

### 2.

Suppose that ~ (0,1). Let Y be independent of X with P(Y = 1) = P(Y = -1) = 1/2. Define the random variable Z be setting Z = XY.

a)ComputeCov(X,Z).

b)Showthat P(Z >= 1) = P(X >= 1). Use this fact to conclude that Z and X are NOT

c)Generalizepart (b) to show that P(Z >= x) = P(X >= x) for every . This implies that ~ (0,1)

### 3.     Statistics统计学作业代写

Let (X, Y) be a point that is uniformly distributed on a square whose corners are (±1, ±1). Determine the distribution(s) of the x- and y-coordinates. Are X and Y independent? Are they uncorrelated?{Exercise 1.3. p19}. See example 1.1.

### 4.

Suppose that X1, X2, and X3 are independent and identically distributed continuous random variables with common density function f(x).

a)Compute P(X1 >X2)

b)Compute P(X1 > X2| X1 >X3)

c)Compute P(X1 > X2| X1 <X2)

Hint: You can answer this problem easily using symmetry.    Statistics统计学作业代写

### 5.Bonus

Let (X, Y, Z) be a point chosen uniformly within the three-dimensional unit sphere. Determine the marginal distributions of (X, Y) and X.

{Exercise 1.1. p18}

Finding the marginal distribution of (X, Y)’ involves a computation similar to Example 1.1. However, finding the marginal distribution of just X is much more frustrating.