20, variances a,a and correlation 4. Let X. Y be normal bivariate r.v. with coefficient p....
If X and Y are two non-independent normal distribution whose joint distributions is bivariate normal with correlation p, what is Var(XY)?
Let X and Y have a bivariate normal distribution with parameters
μX = 10, σ2 X = 9, μY = 15, σ2 Y = 16, and ρ = 0. Find (a) P(13.6
< Y < 17.2). (b) E(Y | x). (c) Var(Y | x). (d) P(13.6 < Y
< 17.2 | X = 9.1).
4.5-8. Let X and Y have a bivariate normal distribution with parameters Ax-10, σ(-9, Ily-15, σǐ_ 16, and ρ O. Find (a) P(13.6< Y < 17.2)...
(a) Show that (Xi, X2) has a bivariate normal distribution with means μ1 , μ2, variances 어 and 05, and correlation coefficient ρ if and only if every linear combination c Xc2X2 has a univariate normal distr bution with mean c1μι-c2μ2, and variance c?σ? + c3- +2c1c2ρσ12, where cı and c2 are real constants, not both equal to zero. (b) Let Yİ = Xi/ởi, i = 1,2. Show that Var(Y-Yo) = 2(1-2).
11. Let the correlation coefficient of X and Y be ρ(X,Y)-N C XY VVar(X)VVar(Y) -p(X, Y). (Y) Show that ρ(-3X,-2Y-a(X, Y) and ρ(X-2Y)
3. Let (X. X2) be standard bivariate normal with p = 3/5. Let (Y.Y2) be the midterm and final exam scores of a randomly selected student. Assume Y1 = 80 +3X1Y2 = 75 + 2X2. Given a student got 90 in the midterm exam, (a) What is the conditional expectation and conditional variance of her final exam score? Hint. Probably easier to reduce the question to (X1, X2) but also (Y1. Y2) is a normal bivariate. (b) What is the...
р 9. If (X,Y) are bivariate normal with E(X) = 20, var(X) = 25, E(Y) = 16, var(Y) = 9, and = 0.7, what is the distribution of Y given X = 30? 3.52 .d.
Suppose (X, Y ) has bivariate
normal distribution, E(X) = E(Y ) = 0,V ar(X) = σX2 , V ar(Y ) =
σY2 and Correl(X, Y ) = ρ. Calculate the conditional expectation
E(X2|Y ).
I. Suppose (X,Y) has bivariate normal distribution, E(X) = E(Y) 0, Var(X)-σ , Var(Y) σ and Correl (X,Y)-p. Calculate the conditional expectation ECKY expectation E(X2Y)
1. Suppose (x, Y) has bivariate normal distribution, E(x) E(Y)- 0, Var(X) σ , Var(Y) σ and Correl(X, Y) p. Calculate the conditional expectation E(X2|Y).
Let X and Y have a bivariate normal distribution with parameters μX = 4, μY = 2, σX = 2, σY = 4, and ρ = 1/2. Find two different lines, a(x) and b(x), parallel to and equidistant from E(Y|x), such that P[a(x) < Y < b(x)|X = x] = 0.6827 for all real x.
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 +...