![(8) 16 pts] For two random variables X and Y, and for constants a,b,c,d R, prove that Var (aX + b) + (cY + d)] = a2 VarlX) + cWarM + 2acCoolx, y In crafting your argument, you are allowed to use any properties of expectations and/or variances that we covered in lecture.](http://img.homeworklib.com/questions/f2085d20-6e57-11ea-b64b-01c17c37bc4c.png?x-oss-process=image/resize,w_560)
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(8) 16 pts] For two random variables X and Y, and for constants a,b,c,d R, prove...
For constants a and b, X and Y are random variables. Please prove that, var(aX +...
For constants a and b, X and Y are random variables. Please prove that, var(aX + bY ) = a 2 var(X) + b 2 var(Y ) + 2abcov(X, Y ) If X and Y are uncorrelated, what will be the results?
Let X and Y be independent identically distributed random variables with means µx and µy respectively. Prove the following. a. E [aX + bY] = aµx + bµy for any constants a and b. b. Var[X2] = E[X2] − E...
Let X and Y be independent identically distributed random variables with means µx and µy respectively. Prove the following. a. E [aX + bY] = aµx + bµy for any constants a and b. b. Var[X2] = E[X2] − E[X]2 c. Var [aX] = a2Var [X] for any constant a. d. Assume for this part only that X and Y are not independent. Then Var [X + Y] = Var[X] + Var[Y] + 2(E [XY] − E [X] E[Y]). e....
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 =
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)
Let X and Y be independent identically distributed random variables with means µx and µy respectively....
Let X and Y be independent identically distributed random variables with means µx and µy respectively. Prove the following. a. E [aX + bY] = aµx + bµy for any constants a and b. b. Var[X2] = E[X2] − E[X]2 c. Var [aX] = a2Var [X] for any constant a. d. Assume for this part only that X and Y are not independent. Then Var [X + Y] = Var[X] + Var[Y] + 2(E [XY] − E [X] E[Y]). e....
Please answer all parts of the question, with all work shown Problem Seven (Properties of Covariance...
Please answer all parts of the question, with all work shown
Problem Seven (Properties of Covariance and Corelation) (A) Prove that you can express Var(aX b,cY d) as for some appropriate constants α, β, and γ. (Note: X and Y can not be assumed to be independent.) (B) Let X" and Y be the standardized versions of the random variables X and Y. Prove that (I suggested this relation in lecture but did not prove it.)
4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) =...
4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) = E(X - Ex)(Y - EY) (a) (2pt) TRUE or FALSE (circle one). E(XY) 0 implies Cov(X, Y) = 0. (b) (4 pt) a, b, c, d are constants. Mark each correct statement ( ) Cov(aX, cY) = ac Cov(X, Y) ( ) Cor(aX + b, cY + d) = ac Cov(X, Y) + bc Cov(X, Y) + da Cov(X, Y) + bd ( )...
= 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
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers,...
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.
2. (10 pts) Random variables X and Y have the following joint PDF: 0.1, if both...
2. (10 pts) Random variables X and Y have the following joint PDF: 0.1, if both 11 and 2S2 Jx( if both Is2 ad Sys; 0, otherwise. (a) Prepare neat, fully labeled sketches of xir (r) (b) Find EKİY=y] and var(X|Y-v). (c) Find E[x (d) Find var(x)using the law of conditional variances.
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 =
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)
Please answer all parts of the question, with all work shown
Problem Seven (Properties of Covariance and Corelation) (A) Prove that you can express Var(aX b,cY d) as for some appropriate constants α, β, and γ. (Note: X and Y can not be assumed to be independent.) (B) Let X" and Y be the standardized versions of the random variables X and Y. Prove that (I suggested this relation in lecture but did not prove it.)
4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) = E(X - Ex)(Y - EY) (a) (2pt) TRUE or FALSE (circle one). E(XY) 0 implies Cov(X, Y) = 0. (b) (4 pt) a, b, c, d are constants. Mark each correct statement ( ) Cov(aX, cY) = ac Cov(X, Y) ( ) Cor(aX + b, cY + d) = ac Cov(X, Y) + bc Cov(X, Y) + da Cov(X, Y) + bd ( )...
= 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
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.
2. (10 pts) Random variables X and Y have the following joint PDF: 0.1, if both 11 and 2S2 Jx( if both Is2 ad Sys; 0, otherwise. (a) Prepare neat, fully labeled sketches of xir (r) (b) Find EKİY=y] and var(X|Y-v). (c) Find E[x (d) Find var(x)using the law of conditional variances.
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