Mean of X1+2X2=0.15+2*0.25=0.65
Standard deviation=sqrt((1*0.1)^2+(2*0.2)^2+2*1*2*0.1*0.2*0.4)=0.449444
Hence, Probability=0.2180
Let ( XvXy) be a bivariate normal vector with returns μι-0.15, μ2 ρ 2. 0.25,01 0.1,...
(a) Let (XI, X2) ~ M2(μι, μ2, σ.σ , p) where μ2-0, and ρ * 1. 41 The polar coordinate transformation is defined by X1- R cos Θ. X2- T2R sin Θ. Show that the joint pdf of R and Θ is given by 0-r < 00, and 0 θ 2π, and that the marginal pdf of Θ is (27)-1 (1-p?)1/2(1-r sin 20)-1, 0 θ 2π. (b) Suppose (X1, X2) has a bivariate normal distribution M2(0,0, σ.σ3.ρ), Ip #1. Show...
Let X = (X1, X2) be a bivariate normal random vector such that Mi = 4,42 = 6,01 = 25, 02 = 16 and p= 0.7. 1. Find P(X2 <5|X1 = 3).
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5 ], = [11] Here -1 <p<1. (a) What is P(X > Y)? (b) Is there a constant c such that X and X +cY are independent?
(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).
3. Let X and Y have a bivariate normal distribution with parameters x -3 , μΥ 10, σ 25, 9, and ρ 3/5. Compute (c) P(7<Y < 16). (d) P(7 < Y < 161X = 2).
3. Let X ~ N2(u, ) be a bivariate Normal random vector, where 6-61-10 X = = | 12 (a) Find the distribution of Y1 = X1 + X2. (b) Let Y2 = X1 +aX2. Find value a such that Yį and Y2 are independent.
QUESTION 6 1 points s Consider the hypothesis test Ho μι-μ2 against H, μι-μ 2 . Suppose that sample sizes are n1-15 and n2-15 and X, 4.5 and X2 -8.0 and that s-4 and s-6.18. Assume that the data are drawn from normal distributions. Compute the p-value of the test statistics. Round your answer to 4 decimal places
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)...
Graybill, 1961]. Let x -(X1, X2) have a bivariate normal distribution with pdf where Q-2x-x1x2 + 4 _ 11x1-5T2 + 19, and k is a constant. Find a constant a such that P(3X1-X2 < a) 0.01.
xercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < 1 . Check that if X Z and Y-: ρΖ+ VI-P" W then the pair (X, Y) has standard bivariate normal distribution with parameter p. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.