4. Suppose X and Y are standard normal random variables. Find an expression for P (X...
Suppose X and Y are standard normal random variables. Find an expression for P (X + 2Y-3) in terms of the standard normal distribution function Φ in two cases: (a) X and Y are independent; (b) X and Y have bivariate normal distribution with correlation p 1/2.
please help me 5. Suppose X and Y are standard normal random variables. Find an expres- sion for P(X - 3Y S1) in terms of the standard normal distribution function In two cases: (i) X and Y are independent (ii) X and Y have bivariate normal distribution with correlation ρ-1/2.
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...
Suppose that the standard normal random variables X and Y are independent. Find P(0 < X<Y). 8 O 1 4T 0 1 8л Ala
10. Let the random variables X ~ NGIX, σ%) and Y ~ Nuy,ơ be jointly continious normal random variables. Now suppose their joint pdf is X and Y are said to have a bivariate normal distribution (a) Given this joint pdf, show that X and Y are independent. (b) The most general form of the pdf for a bivariate normal distribution is What must be true about k for X and Y to be independent bivariate normal random variables? 10....
Exercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < l. Check that if X-Z and Y-p2+ VI-p-W then the pair (X, Y) has standard bivariate normal distribution with parameter ρ. 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.
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.
The random variables Z and W have a bivariate normal dis- tribution with EZ] = E[W] = 0, Var(Z) = Var(W) = 1, and oorrelation ρ E (-1,1). Given that Pl2+ W 1-8413, find the value ofp. Hint: 8413 = φ(1), where φ is the standard normal distribution function.] The random variables Z and W have a bivariate normal dis- tribution with EZ] = E[W] = 0, Var(Z) = Var(W) = 1, and oorrelation ρ E (-1,1). Given that Pl2+...
X and Y be standard normal random variables with correlation ρ. Compute the joint and marginal distributions of X + Y and X-Y . Are X + Y and X-Y independent?
Suppose that X and Y are independent standard normal random variables. Show that U = }(X+Y) and V = 5(X-Y) are also independent standard normal random variables.