Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21...
Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21 and σ22, respectively. Find the correlation of X1 and X1 + X2.
Note that:
The covariance of random variables X; Y is dened by Cov(X; Y ) = E[(X - E(X))(Y - E(Y ))]. The correlation of X; Y
is dened by
Corr(X; Y ) =Cov(X; Y ) / √ Var(X)Var(Y )
Homework Answers
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Let X1 and X2 be independent random
variables with means μ1 and μ2, and variances
σ21...
SupposeX1,...,Xn are independent random samples from a population distributed as N(μ1,σ21) andY1,...,Ym are a independent random...
SupposeX1,...,Xn are independent random samples from a population distributed as N(μ1,σ21) andY1,...,Ym are a independent random samples from N(μ2,σ22). Suppose further that Xi‘s and Yj‘s are independent for any i=1,...,n and j= 1,...,m.a. Determine E[ ̄X+ ̄Y]. b.Determine Var [ ̄X+ ̄Y]. c.What is the distribution of ̄X+ ̄Y?
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and...
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
O. Let X1 and X2 be two random variables, and let Y = (X1 + X2)2....
O. Let X1 and X2 be two random variables, and let Y = (X1 +
X2)2. Suppose that E[Y ] = 25 and that the variance of X1 and X2
are 9 and 16, respectively.
O. Let Xi and X2 be two random variables, and let Y = (X1 X2)2. Suppose that and that the variance of X1 and X2 are 9 and 16, respectively E[Y] = 25 (63) Suppose that both X\ and X2 have mean zero. Then the...
(a) Show that (Xi, X2) has a bivariate normal distribution with means μ1 , μ2, variances...
(a) Show that (Xi, X2) has a bivariate normal distribution with means μ1 , μ2, variances 어 and 05, and correlation coefficient ρ if and only if every linear combination c Xc2X2 has a univariate normal distr bution with mean c1μι-c2μ2, and variance c?σ? + c3- +2c1c2ρσ12, where cı and c2 are real constants, not both equal to zero. (b) Let Yİ = Xi/ởi, i = 1,2. Show that Var(Y-Yo) = 2(1-2).
Proof: If X1 and X2 are independent, then E[(X1-μ1) (X2-μ2)] = 0.
Proof: If X1 and X2 are independent, then E[(X1-μ1) (X2-μ2)] = 0.
Let X1 and X2 be two independent standard normal random variables. Define two new random variables...
Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Y-Xi X2 and Y2- XiBX2. You are not given the constant B but it is known that Cov(Yi, Y2)-0. Find (a) the density of Y (b) Cov(X2, Y2)
Find the standardized test statistic, t, to test the claim that μ1 < μ2. Two samples...
Find the standardized test statistic, t, to test the claim that μ1 < μ2. Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that two populations' variance is the same (σ21= σ22). n1 = 15 n2 = 15 x1 = 25.76 x2 = 28.31 s1 = 2.9 s2 = 2.8
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is...
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
5. Let X1 and X2 be two independent standard normal random variables. Define two new random...
5. Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Yı = X1 + X2 and ½ = X1 + ßX2. You are not given the constant β but it is known that Cov(Yi,Y) = 0. Find (a) the density of Y2 (b) Cov(Xy½),
Exercise 1 (1). X, Y are random variables (r.v.) and a,b,c,d are values. Complete the formulas...
Exercise 1 (1). X, Y are random variables (r.v.) and a,b,c,d are values. Complete the formulas using the expectations E(X), E(Y), variances Var(X), Var(Y) and covariance Cov(X, Y) (a) E(aX c) (b) Var(aX + c (d) Var(aX bY c) (e) The covariance between aX +c and bY +d, that is, Cov(aX +c,bY +d) f) The correlation between X, Y that is, Corr(X,Y (g) The correlation between aX +c and bY +d, that is, Corr(aX + c, bY +d)
O. Let X1 and X2 be two random variables, and let Y = (X1 +
X2)2. Suppose that E[Y ] = 25 and that the variance of X1 and X2
are 9 and 16, respectively.
O. Let Xi and X2 be two random variables, and let Y = (X1 X2)2. Suppose that and that the variance of X1 and X2 are 9 and 16, respectively E[Y] = 25 (63) Suppose that both X\ and X2 have mean zero. Then the...
(a) Show that (Xi, X2) has a bivariate normal distribution with means μ1 , μ2, variances 어 and 05, and correlation coefficient ρ if and only if every linear combination c Xc2X2 has a univariate normal distr bution with mean c1μι-c2μ2, and variance c?σ? + c3- +2c1c2ρσ12, where cı and c2 are real constants, not both equal to zero. (b) Let Yİ = Xi/ởi, i = 1,2. Show that Var(Y-Yo) = 2(1-2).
Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Y-Xi X2 and Y2- XiBX2. You are not given the constant B but it is known that Cov(Yi, Y2)-0. Find (a) the density of Y (b) Cov(X2, Y2)
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
5. Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Yı = X1 + X2 and ½ = X1 + ßX2. You are not given the constant β but it is known that Cov(Yi,Y) = 0. Find (a) the density of Y2 (b) Cov(Xy½),
Exercise 1 (1). X, Y are random variables (r.v.) and a,b,c,d are values. Complete the formulas using the expectations E(X), E(Y), variances Var(X), Var(Y) and covariance Cov(X, Y) (a) E(aX c) (b) Var(aX + c (d) Var(aX bY c) (e) The covariance between aX +c and bY +d, that is, Cov(aX +c,bY +d) f) The correlation between X, Y that is, Corr(X,Y (g) The correlation between aX +c and bY +d, that is, Corr(aX + c, bY +d)
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