4. Suppose that X and Y are random variables with E(X) = 2, E(Y) = 1....
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 two random variables Y1 and Y2 have the following quantities: E(Y) = 3, E(Y/2) = 18, E(Y2) = 5, E(Y22) = 29, E(Y1Y2) = 11 Find the correlation coefficient of Y1 and Y2. That is to find the value of Corr(Y 1, Y2) -4.0000 0.6667 O -0.1111 -0.6667
= 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
4. Suppose X and Y are independent random variables with the same probability distribution, given by the cumulative distribution function if t 2 1 if t < 1 F(t)= 1 -t-3 (a) (10 points) Compute E(X). (b)(10 points) Compute E(XY). Chr
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...
1. Suppose X and Y are continuous random variables with joint pdf f(x,y) 4(z-xy) if = 0 < x < 1 and 0 < y < 1, and zero otherwise. (a) Find E(XY) b) Find E(X-Y) (c) Find Var(X - Y) (d) What is E(Y)?
The following relates to Problems 26 - 27. Let X, Y be random variables and b a number. Problem 26: Find E (Y – bX)21 [1] E(X)b2 – 2E(XY)b+E(Y2); [2] E(X2)62 – E(Y); [3] -2E(XY)b+E(Y); [4] E(Y2); 151 E(X2)62 [6] Problem 27: Find b that minimizes E [(Y – bx)2] [1] E(YP); [2] E(X) – E(Y); [3] –2E(XY) + E(Y2); [4] ; [5]
13. Consider the random variables X and Y with the following expectations: E(X)= 2, E(Y)=1 E(X²)=15, E(Y2)=9, E(XY)=1. Let U = X + 2Y, V = 3X - Y and calculate the covariance of U and V.
2. Suppose X and Y are continuous random variables with joint density function f(x, y) = 1x2 ye-xy for 1 < x < 2 and 0 < y < oo otherwise a. Calculate the (marginal) densities of X and Y. b. Calculate E[X] and E[Y]. c. Calculate Cov(X,Y).
Suppose X and Y are continuous random variables with joint density function 1 + xy 9 fx,y(2, y) = 4 [2] < 1, [y] < 1 otherwise 0, (1) (4 pts) Find the marginal density function for X and Y separately. (2) (2 pts) Are X and Y independent? Verify your answer. (3) (9 pts) Are X2 and Y2 independent? Verify your answer.