9.28 Let X and Y be independent standard normal random variables. Find the mgf of X2...
Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Y-Xi X2 and Y2- XiBX2. You are not given the constant B but it is known that Cov(Yi, Y2)-0. Find (a) the density of Y (b) Cov(X2, Y2)
5. Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Yı = X1 + X2 and ½ = X1 + ßX2. You are not given the constant β but it is known that Cov(Yi,Y) = 0. Find (a) the density of Y2 (b) Cov(Xy½),
3, Let X, X2,X, be independent random variables such that Xi~N(?) a. Find the distribution of Y= a1X1+azX2+ i.(Hint: The MGF of Xi is Mx, (t) et+(1/2)t) + anXn +b where a, 0 for at least one b. Assume = 2 =n= u and of- a= (X-)/(0/n) ? Explain. a. What is the distribution of The Sqve o tubat num c. What is the distribution of [(X-4)/(0/Vm? Explain.
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4. Let Xi and X2 be two independent standard normal random variables Define two new random variables as follows: Yǐ = X1 +X2 and Y2 = X1 +ßX2. Y t ß but it is known that Cor(Y,Y)-0. Find ou are not given the const an (i) The density of Y2 . (ii) Cov(X2, Y2). (your answers shouldn't involve β)
1. Let X1, X2, , Xn be independent Normal μ, σ2) random variables. Let y,-n Σ_lx, denote a sequence of random variables (a) Find E(y,) and Var(y,) for all n in terms of μ and σ2. (b) Find the PDF for Yn for alln. (c) Find the MGF for Yn for all n.
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12 X2 = 321 - 22 +11. (a) Find the joint density function of (X1, X2). (b) Find the covariance of X1 and X2. Now let Y1 = X1 + 4X2 +3 Y, = -2X2 +6X2 +5 (a) Find the joint density function of (Y1, Y). (b) Find the covariance of Yi and Y2.
Let Y.Y2, ,Yn be independent standard normal random variables. That is, Y i-1,... ,n, are iid N(0, 1) random variables. 25 a) Find the distribution of Σ 1 Y2 b) Let Wn Y?. Does Wn converge in probability to some constant? If so, what is the value of the constant?
2. Let X and Y be independent, standard normal random variables. Find the joint pdf of U = 2X +Y and V = X-Y. Determine if U and V are independent. Justify.
Let Z1, Z2,.., Zn be independent Normal(0,1) random variables (a) Find the MGF for Z for all i (b) Find the MGF for (c) If n is even, find the PDF for Σ
Let Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n