d)
here c1=required probability in question (a) and c2=required probability in question (b)
Question 5 - Even More Fun With Bivariate Normal Distributions Let X and Y be independent normall...
If X and Y are two non-independent normal distribution whose joint distributions is bivariate normal with correlation p, what is Var(XY)?
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...
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).
Let X be a standard normal distribution. Let ξ be another random variable, independent of X, which can take only two possible values, say -1 and 1. Moreover, assume that Ele] = 0. ( . (b) Find COV(x,Y). (c) Are X and Y independent? (d) Is the pair (X,Y) bivariate normal? a) Find the distribution of Y -£X Let X be a standard normal distribution. Let ξ be another random variable, independent of X, which can take only two possible...
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?
Let Yi, Ys,.., Y's be a random sample of size 5 from a normal distribution mean 0 and standard deviation 1 and let-3x /5 . Let Y6 be another independent observation from the same distribution. Find the distributions of the following random variables i-1 2(572 +Y) (b) WW Let Yi, Ys,.., Y's be a random sample of size 5 from a normal distribution mean 0 and standard deviation 1 and let-3x /5 . Let Y6 be another independent observation from...
5. Suppose that X and Y are independent with distributions N(0,0) and N(0,02), respectively. Let Z=X+Y. Also, let W = 02X – oʻY. Prove that Z and W are uncorrelated.
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
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).
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....