variable X and Y is independent. How to prove that E(XY)=E(X)E(Y) use intgration. There is some kind double integration and I dont understand it.
variable X and Y is independent. How to prove that E(XY)=E(X)E(Y) use intgration. There is some...
Q1/ Use algebraic manipulation to prove that (xy)(x y) x
Let xi be independent. E(xi)=0. Var(xi)= sigma ^2
Cov(x,y) = E(XY) - ExEy
Use this fact and apply it to this example ! Do not use
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Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y) Var(X).
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 =
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...
Problem #1 Y1(x)= x and Y2(x)=e* are linearly independent solution of the homogeneous equation: (x-1)y"-xy'+y = 0 Find a particular solution of (x-1) y”-xy’+y = (x-1)} e2x
Let X, Y E Mn (R). Prove that XY = XY_if and only if there exists an invertible matrix Z so that X = Z In and Y = Z1 + In. Hint: the trace is not involve at all in this problem _
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....
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ). (b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer. (c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ. (d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify...
9. Some TRUE/FALSE questions. The RVs X and Y must be independent if... (a) f(y|x) = fy(x) for all X. (b) Cov(X,Y) = 0. (c) f(x, y) = cx, for 0 < x < y2 < 1. (d) E[XY] = E[X] · E[Y]. (e) f (x, y) = cx- (1 + y2), for 0 < x <1,0 < y < 2.