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)
Suppose (X,Y) follows a trinomial distribution (5, 1/3, 1/4). a. Find E(X) b. Find E(Y) c....
1. Suppose you have two random variables, X and Y with joint distribution given by the following tables So, for example, the probability that Y o,x - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),J(Y). (b) Find the conditional distribution (pmf) of Y give X, denoted f(YX). (c) Find the expected values of X and Y, EX), E(Y). (d) Find the variances of X and Y, Var(X),Var(Y). (e)...
Suppose X and Y are jointly distributed with density \ a. Find c. (b) Find the marginal distribution of X and Y. (c) Find P(X > 2). (d) Find (E[X2 ]). (e) Find the conditional distribution of Y, given that X = 1. (f) E[X], E[Y], E[XY], Cov(X,Y) and ρXY f(x,y)ce-(=/2+y/4) 0<y<I < 0 otherwise 0 f(x,y)ce-(=/2+y/4) 0
The following relates to Problems 21 and 22. Let X ~ NĢi 1, σ2-1), Y ~ NĢı = 2,02-9) and ρχ.Y = 0.5 (recall that ρΧΥ stands for the correlation coefficient of X and Y) Problem 21: Find COV(X, Y) and Var(X +Y) 1 COV(X, Y) 1.5 and Var(XY)-15; [2] COV(X, Y) 3 and Var(X+Y)-7; 3 COV(X,Y) 3 and Var(X + Y) 10: 4] COV(X,Y) 1.5 and Var(X + Y)-7; [5] cov (X, Y) = 1.5 and Var (X +...
Let X and Y have the following joint distribution: X/Y 0 1 2 0 5/50 8/50 1/50 2 10/50 1/50 5/50 4 10/50 10/50 0 Further, suppose σx = √(1664/625), σy = √(3111/2500) a) Find Cov(X,Y) b) Find p(X,Y) c) Find Cov(1-X, 10+Y) d) p(1-X, 10+Y), Hint: use c and find Var[1-X], Var[10+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 =
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
15 Let X and Y have a trinomial distribution with n = 8, P1 = 0.4 and P2 = 0.1 f (x,y) = 8! 10.4*0.190.58---8,0 < x + y < 8,2 € N, Y EN x!y! (8 - x - y)! (a) Find E (Y|X = x), Var (Y|X = x) (b) Compute E (XY)
17. From the following joint probability distribution of X and Y,f (a) P(X= Y], P [X > Y] (b) the distribution of X+ Y (c) E(X), E(Y), Var(X), Var(Y) (d) Cov(X, Y), Corr(X, Y) nd: 0 .3 .25 .05 .2 19 T
2. Suppose the variables Yi and Y have the following properties EQİ)-4, Var(h)-19, E(Y )-6.5, Var(Ya)-5.25, E(Y3%)-30 Calculate the following; please show the underlying work a) (3 pts) Cov(, ) b) (3 pts) Cov(41, 3%) c) (3 pts) Cov(41.5-½) (6 pts) Find the correlation coefficient between 1 + 3, and 3-2%
4. Suppose X and Y has joint density f(x, y) = 2 for () < x <y<1. (a) Find P(Y - X > 2). (b) Find the marginal densities of X and Y. (c) Find E(X), E(Y), Var(X), Var(Y), Cov(X,Y)