X ~ Ber(6, 0.5) and Y ~ Ber(2, 0.25) are independent. Determine Z = X - 2Y, and use MGF to calculate E(Z) and Var(Z).
X ~ Ber(6, 0.5) and Y ~ Ber(2, 0.25) are independent. Determine Z = X -...
You are given three independent random variables X, Y, and Z, all distributed exponentially, such that the hazard rate of X is Ax, the hazard rate of Y is ly, and the mean of Z is 4. You are also given that E (Y + Z) = Var (Y - X) and Var (X + Y + 2) = 3E (2Y + Z). Find dy - dx. Possible Answers A -0.05 D 10.05 20.09
X,Y, and Z are random variables. Var(X) = 2, Var(Y) = 1, Var(Z) = 5, Cov(X,Y) = 3, Cov(X, Z) = -2, Cov(Y,Z) = 7. Determine Var(3X – 2Y - 2+10)
Q2: Suppose that X-N(O, 1), U-N(O, 0.25), Y 3- 2X and Z following questions. 2 X +U. Please answer the Compute E(Y), E(Z), Var(Y) and Var(Z). What are the distributions of Y and Z? Using R, draw 50 independent realizations of X and U. Using those values, create 50 realizations of Y and Z. (NOTE: set the seed for random number generation in R. Before your code type set.seed 123))
X and z are independent. Var(x)=1 ; Var(z) = sigma ^2. E(z)= 0 and y= ax+b+z I) cov(x,y)= ? ii) corr(x,y)=? dependent Varvane 2.
Obtain E(Z|X), Var(Z|X) and verify that E(E(Z|X)) =E(Z), Var(E(Z|X))+E(Var(Z|X)) =Var(Z) 3. Let X, Y be independent Exponential (1) random variables. Define 1, if X Y<2 Obtain E (Z|X), Var(ZX) and verify that E(E(Zx)) E(Z), Var(E(Z|X))+E(Var(Z|X)) - Var(Z)
3. (4 points) The random variables X and Y are independent and have moment generating functions Find Var(X).x (t) =er-2t and Mr(t)=e3t2+tid t a) Find MGF of Z Find Var(Z). Find joint MGF of X and Z, i.e. Mxz(t1,t2) 2X-Y c) d)
4. Consider two independent random variables X and Y, such that E[X] = 1 E[Y] = 2 var(X) = 2 var(Y) = 1 Let Z = X-Y 2 (a) Calculate E[2] (b) Calculate var(Z). 3
Suppose that X, Y and Z are all independent of each other, with the following distributions: X Poisson(1) Y ~ Gamma(a,b) ZN(0,1) Define A as the sum: A = X+Y+Z a What is E[A]? b What is the MGF of A? (you don't need to re-derive the individual mgfs) c Use mA(t) to find E[A] (should match part a)
Question 3 (4101) Suppose that X, Y and Z are all independent of each other, with the following distributions: X Poisson (1) Y Gamma(,b) Z~ N(0,1) Define A as the sum: A = X+Y+Z a What is E[A]? b What is the MGF of A? (you don't need to re-derive the individual mgfs) c Use ma(t) to find E[A] (should match part a)
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).