Let X and Y be independent random variables with X = N(0, 1) and Y = Exp(1). Find E( |X| (Y + 1)^2 ).
Let X and Y be independent random variables with X = N(0, 1) and Y = Exp(1). Find E( |X| (Y + 1)^2 ).
2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution 2) Let X,..X, be ii.d. N(O, 1) random variables. Define U- Find the limiting distribution of Zn (Hint: Recall that if X and Y are independent N(0, 1) random variables, then has a Cauchy distribution
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b) 3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = Oy = 5. Then Var(2x+3Y) = 1. True False
Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = oy = 5. Then Var(2x +3Y) = 1. True False
Let X and Y be two independent random variables such that E(X) = E(Y) = u but og and Oy are unequal. We define another random variable Z as the weighted average of the random variables X and Y, as Z = 0X + (1 - 0)Y where 0 is a scalar and 0 = 0 < 1. 1. Find the expected value of Z , E(Z), as a function of u . 2. Find in terms of Oy and...
Let X and Y be two independent random variables with X =d R(0, 2) and Y =d exp(1). (a) Use the convolution formula to calculate the probability density function of W =X+Y. (b) Derive the probability density function of U = XY .
2) Let X and Y be independent exponential random variables with means E[X] = 0 and EY = 28. 1 1 f(310) = -X/0 e x > 0, f(y|0) = e-4/20 y > 0 0 24 a) Show that the likelihood function can be written as (2 points) L(0) = e-3(x+3) 202 b) Find the MLE ô of 0. (5 points)
Let X and Y be independent random variables with pdf 2-y , 0sys2 2 f(x) 0, otherwise 0, otherwise ) Find E(XY) b) Find Var (2X+3Y)
Let X and Y be independent variables with X ~ EXP(mu(x)) and Y ~ EXP (mu(y)), where mu(x) = 1 and mu(y) = 1/2. Write explicit integral expressions for each of the following, without computing the values. P(Y < X)
2. Let X and Y be two independent discrete random variables with the probability mass functions PX- = i) = (e-1)e-i and P(Y = j-11' for i,j = 1, 2, Let {Uni2 1} of i.i.d. uniform random variables on [0, 1]. Assume the sequence {U i independent of X and Y. Define M-max(UhUn Ud. Find the distribution