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.
Let X and Y have a bivariate normal distribution with parameters μX = 4, μY =...
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)...
μx= 10, σx =5, μy=8, σy=4, ρ=0 X and Y come from normal distributions Calculate the probability that X > Y. Z = 3X + 10 Calculate the probability that Z > 45.
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).
If X and Y have a bivariate normal distribution with parameters mean1,mean2, variance1, variance2and P show that Z = aX + bY + c is N(a.mean1 + b.mean2 + c, variance1.variance2 + 2abp.variance1.variance2 + b^2.variance2, where a, b, and c are constants. Hint: Use the m.g.f. M(t1 t2) of X and Y to find the m.g.f. of Z.
9. Let X and Y be two random variables. Suppose that σ = 4, and σ -9. If we know that the two random variables Z-2X?Y and W = X + Y are independent, find Cov(X, Y) and ρ(X,Y). 10. Let X and Y be bivariate normal random variables with parameters μェー0, σ, 1,Hy- 1, ơv = 2, and ρ = _ .5. Find P(X + 2Y < 3) . Find Cov(X-Y, X + 2Y) 11. Let X and Y...
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?
The observations make up the population of the variable X: X1 = 2, X2 = 3, X3 = 4 a. Find the population mean of X, μX. b. Find the population standard deviation of X, σX. Suppose that the variable Y is defined as follows: Y = (X – μX) / σX c. Calculate Y1, Y2, and Y3. d. Find the mean of Y, μY e. Find the standard deviation of Y, σY.
17. Suppose that (X,Y) has a bivariate normal distribu- zion with parameters diy, x, 0y.p. io show that (2 , 4") has a bivariate normal distri- bution with parameters 0, 1,0.1.p. b) What is the joint distribution of (aX + b,cY + 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)
20, variances a,a and correlation 4. Let X. Y be normal bivariate r.v. with coefficient p. a) Write what are E (X|Y), var (X|Y)? b) Show that σi + σισ E (XXY) afo(-p) +2pa 102+0 var (XXY 2) Hint. (X, XY)is normal bivariate: apply a).