![20. For X let E(X)-0 and sd(x)-2, and for Y let E(Y)--1 and sd(Y)-4. Find: (a) E(X-Y) and E (X Y). (b) Var(X- Y) and Var(X+ Y) if X and Y are independent. (c) EGX+ 흘 Y) and Var(1X+] Y) İf X and Y are independent. (d) Repeat (b) if, instead of independence, Cov(X, Y)- 1. soY is VarY larger](http://img.homeworklib.com/questions/ad5b7cc0-f5e4-11eb-b62f-91cfad5cf1ae.png?x-oss-process=image/resize,w_560)
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20. For X let E(X)-0 and sd(x)-2, and for Y let E(Y)--1 and sd(Y)-4. Find: (a)...
Let the random variable X and Y have joint pdf f(x,y)=4/7(x2 +3y2), 0<x<1, 0<y<1 a. find...
Let the random variable X and Y have joint pdf f(x,y)=4/7(x2 +3y2), 0<x<1, 0<y<1 a. find E(X) and E(Y) b. find Var(X) and Var(Y) c. find Cov (X,Y)
Let X and Y have the following joint distribution: X/Y 0 1 2 0 5/50 8/50...
Let X and Y have the following joint distribution: X/Y 0 1 2 0 5/50 8/50 1/50 2 10/50 1/50 5/50 4 10/50 10/50 0 Further, suppose σx = √(1664/625), σy = √(3111/2500) a) Find Cov(X,Y) b) Find p(X,Y) c) Find Cov(1-X, 10+Y) d) p(1-X, 10+Y), Hint: use c and find Var[1-X], Var[10+Y]
4. Assume X ~ Uniform(0, 1) and let Y = 2X+1 and Z = X2 +...
4. Assume X ~ Uniform(0, 1) and let Y = 2X+1 and Z = X2 + 1. (a) Find Cov(X,Y), Var(X+Y), Var(X - Y) and Corr(X,Y). (b) Find Cov(X, Z), Var(X + Z), Var(X – Z) and Corr(X, Z).
Exercise 2 (2). Let X and ε be independent normally distributed random variables such that X∼N(5,4)andε∼N(0,9).LetY...
Exercise 2 (2). Let X and ε be independent normally distributed random variables such that X∼N(5,4)andε∼N(0,9).LetY bearandomvariablegivenbyY =1+2X+ε.Compute: (a) E(Y ) (b) Var(Y ) (c) Cov(X, Y ) (d) Corr(X, Y ) (e) What is the value of the ratio Cov(X, Y )/Var(X) ? (f) If Y = 1 + 3X + ε instead, what would be the value of Cov(X, Y )/Var(X) ? (g) If Y = 1 + 7X + ε instead, what would be the value of...
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distributio...
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distribution of X given Y-0. (i.e., Pxjy (xY-0)) (c) Compute EXIY-01 and Var(X)Y = 0). (d) Find the covariance between X and Y. (i.e., Cov(X, Y)) (e) Are X and Y independent? Justify your answer.
(20 points) Consider the following joint distribution of X and...
1. (20 pts) RVs X and Y have joint density function 22 f(x, y) =(0 if O <z<1 and 0<y<2 īf 0 < x < 1 and 0 < y < 2 otherwise (a) Find E(X), V(X), E(Y), and V(Y). (b) Find t...
1. (20 pts) RVs X and Y have joint density function 22 f(x, y) =(0 if O <z<1 and 0<y<2 īf 0 < x < 1 and 0 < y < 2 otherwise (a) Find E(X), V(X), E(Y), and V(Y). (b) Find the covariance cov(X,Y) and the associated correlation ρ (c) Find the marginal densities fx and fy. (Be sure to say where they're nonzero.) (d) Find E(X | Y = 1.5). (e) Are X and Y independent? Give two...
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) =...
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
Suppose (X,Y) follows a trinomial distribution (5, 1/3, 1/4). a. Find E(X) b. Find E(Y) c....
Suppose (X,Y) follows a trinomial distribution (5, 1/3, 1/4). a. Find E(X) b. Find E(Y) c. Find Var(X) d. Find Var(Y) e. Find Cov (X,Y) f. Find p (correlation coefficient)
= Var(X) and σ, 1. Let X and Y be random variables, with μx = E(X),...
= Var(X) and σ, 1. Let X and Y be random variables, with μx = E(X), μY = E(Y), Var(Y). (1) If a, b, c and d are fixed real numbers, (a) show Cov (aX + b, cY + d) = ac Cov(X, Y). (b) show Corr(aX + b, cY +d) pxy for a > 0 and c> O
Assume X, Y are independent with EX = 1 EY = 2 Var(X) = 22 Var(Y)...
Assume X, Y are independent with EX = 1 EY = 2 Var(X) = 22 Var(Y) = 32 Let U = 2X + Y and V = 2X – Y. (a) Find E(U) and E(V). (b) Find Var(U) and Var(V). (c) Find Cov(U,V).
4. Assume X ~ Uniform(0, 1) and let Y = 2X+1 and Z = X2 + 1. (a) Find Cov(X,Y), Var(X+Y), Var(X - Y) and Corr(X,Y). (b) Find Cov(X, Z), Var(X + Z), Var(X – Z) and Corr(X, Z).
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distribution of X given Y-0. (i.e., Pxjy (xY-0)) (c) Compute EXIY-01 and Var(X)Y = 0). (d) Find the covariance between X and Y. (i.e., Cov(X, Y)) (e) Are X and Y independent? Justify your answer.
(20 points) Consider the following joint distribution of X and...
1. (20 pts) RVs X and Y have joint density function 22 f(x, y) =(0 if O <z<1 and 0<y<2 īf 0 < x < 1 and 0 < y < 2 otherwise (a) Find E(X), V(X), E(Y), and V(Y). (b) Find the covariance cov(X,Y) and the associated correlation ρ (c) Find the marginal densities fx and fy. (Be sure to say where they're nonzero.) (d) Find E(X | Y = 1.5). (e) Are X and Y independent? Give two...
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
= Var(X) and σ, 1. Let X and Y be random variables, with μx = E(X), μY = E(Y), Var(Y). (1) If a, b, c and d are fixed real numbers, (a) show Cov (aX + b, cY + d) = ac Cov(X, Y). (b) show Corr(aX + b, cY +d) pxy for a > 0 and c> O
Assume X, Y are independent with EX = 1 EY = 2 Var(X) = 22 Var(Y) = 32 Let U = 2X + Y and V = 2X – Y. (a) Find E(U) and E(V). (b) Find Var(U) and Var(V). (c) Find Cov(U,V).
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