# 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 =

TOPIC:Properties of expectation,variance and covariance.     ##### Add Answer of: Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) =...
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Prove:(1) E(X+Y)=E(x)+E(y)(2) Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X; Y ).(3)If X and Y are independent, i.e., f(x; y) = fx(x)fy(y), then Cov(X; Y ) = 0.

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Please help me please. They are due on WednesdayFor two continuous random variables X and Ywith a joint density functionf(x;y), prove(1) E[X +Y ] = E[X] +E[Y ].(2) Var[X +Y ] = Var[X] +Var[Y] + 2Cov(X; Y ).(3) If X andY are independent, i.e., f(x;y) = fX(x)fY (y), thenCov(X; Y) = 0.Explain thatthereverse may not be true, i.e., if Cov(X; Y ) =0, Xand Y may not beindependent.

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• ### X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y )... X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y ) P((X x})P({Y y)) then show that E(XY) = E(X)E(Y), i.e. if two random variables are independent, then show that they are uncorrelated. Is the reverse true? Prove or disprove (c) The moment generating function of a random variable Z is defined as ΨΖφ : Eez) Now if X and Y are independent random variables then show that Also, if ΨΧ(t)-(λ- (d) Show the conditional...

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• ### 2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density f... 2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density function. Use f(x.y) with that value of k as the joint probability density function of X, Y in parts (b),(c).(d),(e (b) Find the probability density functions of X and Y. (c) Find the expected values of X, Y and XY (d) Compute the covariance Cov(X,Y) of X...

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