b. Suppose , and Xy are independent with「(a, 1) and r 1,1) distributions. Let Y =...
a. Derive the distribution of the sample mean X of independent Xi,... , Xn where, What x, ~「(α, β). Find a transformation of X that follows a X2 distribution. are the degrees of freedom of this transformation? b. Suppose l and X2 are independent with I(a, 1) and I(e+ , 1) distributions. Let Y = 2VNX2. Find EY and var(Y).
b. Suppose ~ Γ(α, β), with α > 0, β > 0 and let Y-eu. Find the probability density function of Y Find EY and var(Y)
Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y) Var(X).
Assume X, Y are independent with EX = 1 EY = 2 Var(X) = 22 Var(Y) = 32 Let U = 2X + Y and V = 2X – Y. (a) Find E(U) and E(V). (b) Find Var(U) and Var(V). (c) Find Cov(U,V).
10. Let X and Y have a discrete joint distribution with if (x,y) = (-1,1) P(X = 2, Y = y) = { = ; if x=y=0 = 0, elsewhere Find (a) the conditional distribution of Y given X = -1. (b) show that X and Y are uncorrelated but not independent. (C) Find the marginal distributions of X and Y.
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.
If X and Y are two non-independent normal distribution whose joint distributions is bivariate normal with correlation p, what is Var(XY)?
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
probability course
01) 6 and Let X and Y be two independent random variables. Suppose that we know Var(2X-Y) Var(X+ 2Y) 9, Find Var(X) and Var(Y).
6. Let B = {1+1,-1,1+2+r} and C = {1 - 1,1+1,4 – 2. – 2?} be bases for P2(R). Find the change of coordinates matrix cPp from B-coordinates to C-coordinates.