Suppose X and Y V(X) = 3 and V(Y ) = 5. Find:
(a) V(X + Y ) given that X, Y are independent
(b) V(3X + 4) given that X, Y are independent
(c) V(X + X) given that E(X · Y ) = −1
(d) V(X + 3Y ) given that E(X · Y ) = 0
Suppose X and Y have the joint pdf f (x, y) = 3y, 0 < y < 1, y − 1 < x < 1 − y 0 otherwise a) Give an expression for P (X > Y ). b) Find the marginal pdfs for Y . c) Find the conditional pdf of X given Y = y, where 0 < y < 1. d) Give an expression for E[XY ]. e) Are X and Y independent?
1. Suppose that E(X) E(Y) E(Z) 2 Y and Z are independent, Cov(X, Y) V(X) V(Z) 4, V(Y) = 3 Let U X 3Y +Z and W = 2X + Y + Z 1, and Cov(X, Z) = -1 Compute E(U) and V (U) b. Compute Cov(U, W). а.
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y < (c) Find the marginal pdf of X. (d) Find the conditional pdf of Y given that X = x. (e) Find E[Y IX x (f) Find E[E[YX]]. (g) Find Cov(X, Y) (h) Are X and Y independent?
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y
Find a basis B for the domain of T such that the matrix of T(x, y) = (3x + 3y, 3x + 3y) relative to B is diagonal. a B = {(1, -1), (1, 1)} b B = {(1,0), (0, 1); c. B = {(1, 0), (1, 1); d. B = {(1, -1), (1, 0)) e B = {(0, 1), (1, 1);
5. Suppose that the line tangent to the graph of y = f(x) at x = 3 passes through the points (-2,3) and (4, -1). What is the equation of the line tangent to f(x) at x = 3? (1) y=x+ (3) y = -x +5 (1) y= {x +3 (2) y 6. Let f(x) = 7 sin(x + 7) + cos2x. Compute f(13)(O). (1) 8 (2) 7 (3) -8 (1) -7 ----- 7. Differentiate f(x) - 1.6+3 Vã+1 (1)...
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)
1. Suppose the joint density of X and Y is given by f(x,y) = 6e-3x-2y, if 0 < x < inf., 0 < y < inf, 0 elsewhere. Part A, Find P( X < 2Y) Part B, Find Cov(X,Y) Part C, Suppose X and Y have joint density given by f(x,y) = 24xy, when 0<= x <=1, 0 <= y <=1, 0 <= x+y <=1, and 0 elsewhere. Are X and Y independent or dependent random variables? why?
Q3. . Suppose that joint probability function of X and Y is given by | 1/7, z = 5, y = 0 Px,y(, ) 0, otherwise. a. Find the marginal distribution of X and Y b. Find E(X|y = 4] c. Compute Cov(X, Y). d. Are X, Y independent? justify e. Compute E[XY0or4] f. Find px(8) and P(Y-4X-8).
Use the transformation u = 3x + y, v=x + 3y to evaluate the given integral for the region R bounded by the lines y = - 3x + 1, y= - 3x + 3, y= - = X, and y=- -x + 2. ne lines y = – 3x+1, y = – 3x+3, y=-3x, and y=-**+2. 3 Siſ(3?+ 16 +3%) dx ay SJ (3x? + 10x9 +35) dx dy=0 (Simplify your answer.)
Suppose X is a random variable taking on possible values 1,2,3 with respective probabilities.4, .5, and .1. Y is a random variable independent from X taking on possible values 2,3,4 with respective probabilities .3,.3, and 4. Use R to determine the following. a) Find the probability P(X*Y = 4) b) Find the expected value of X. c) Find the standard deviation of X. d) Find the expected value of Y. e) Find the standard deviation of Y. f) Find the...