(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y...
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
5. Let X be a non-central x2 (5, A) random variable, and Y, independent of X, be a x(4) randonm variable. (a). Derive the moment generating function of 2X - 1, and find its mean and variance. (b). Find the mean of W =
Problem 4 Suppose X1, ..., Xn ~ f(x) independently. Let u = E(Xi) and o2 = Var(Xi). Let X Xi/n. (1) Calculate E(X) and Var(X) (2) Explain that X -> u as n -> co. What is the shape of the density of X? (3) Let XiBernoulli(p), calculate u and a2 in terms of p. (4) Continue from (3), explain that X is the frequency of heads. Calculate E(X) and Var(X). Explain that X -> p. What is the shape...
Suppose X and Y are jointly continuous random variables with joint density function Let U = 2X − Y and V = 2X + Y (i). What is the joint density function of U and V ? (ii). Calculate Var(U |V ). 1. Suppose X and Y are jointly continuous random variables with join density function Lei otherwise Let U = 2X-Y and V = 2X + y (i). What is the joint density function of U and V? (ii)....
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo 5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
2. Let Mx(t) = 1c' + 2t?c". Find the following: (b) Var(X). (c) If Y = X-2, show that the moment-generating function of Y is e-2tMx(t). (d) If W = 3X, show that the moment-generating function of W is MX(3). 7/3,5/9
Let Mx(t) = 릉et + 24 + 킁e3. Find the following: (a) E(X) (b) Var(X) (c) If Y - X - 2, find the moment generating function of Y (d) If W -3X, find the moment generating function of W
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n 9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...
1. Suppose X and Y are jointly continuous random variables with joint density function otherwise Let U 2X-Y and V-2X +Y (i). What is the joint density function of U and V? (ii). Caleulate Var(UV)
1. Suppose X and Y are jointly continuous random variables with joint density function otherwise Let u=2x-Yand, V = 2X + Y (i). What is the joint density function of U and V? (ii). Calculate Var(UIV).