6. Suppose the covariances between Xi and X2 is 3, between Xi and Xs is 2, and between X2 and X i...
3. You may use this fact throughout: For any scalars a, a2,a3 and random variables .X2, X3: (a) If Cov (Xi, X2) Cov (X2, X3)-p, Cov (Xi, X3)-p and Var(X1,2,3, then write the 3 x 3 covariance matrix of the random vector X = (X1,X2,X3). (b) Compute Var(Xi X2+X3) when p 0.6. (e) Mark is interested in forecasting X using the linear predictor &bbX He realizes the forecast error is X - X X bX2 -bX and a great way...
Let X- (Xi, X2,X3) be an absolutely continuous random vector with the joint probability density function elsewhere. Calculate 1. the probability of the event A -(Xs 3. the probability density function xx (,s) of the (XX)-marginal 4. the probability density function fx, () of the Xi-marginal, and the probability density function fx (r3) of the X3-marginal 5. Are Xi and X independent random variables? 6. E(Xi) and Var(X) 8. the covariance cov(Xi, X3) of Xi and X,3 9. Which elements...
Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 3 1 0.5 1 E [X]-μ | 0 | and var(X)=Σー| 0 0.5 | com pute: 2 c) var(X2-X3 (d) var(X2 + X3) (e) cov(4X2 +X1,3Xi - 2X3)
Given three random variables Given three random variables Xi, X2, and X such that X[Xi X2 Xa, 2 1 0.5 1 (a) EX, + c) var(X2- X3 (d) var(X2 + X3) (e) cov(4X2 +X1,3Xi - 2X3)
1. (20 points) Let X (Xi, X, Xs) be a real random vector, where X, are identically dis- tributed and independent (ii.d.) zero-mean Gaussian real random variables. Consider the random vector Y given by where A is a 3 x 3 real matrix and b is a 3 x 1 real vector. Justify all your answers. (a) Find the covariance matrix Cx of x. (b) Find the mean vector EY] of Y (c) Express the covariance matrix Cy of Y...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-: μί ald Var(Xi)-σ? where i-| , 2, , n. If ai, aam., an are constants. (i) Write down expression for (i) E{E:-aiX.) and (ii) Var(Σ-lai%). (i) Rewrite the expression if X,'s are not independent.
5. [22] If Xi, X2, and X3 are independent random variables with E(XI) 4, E(X2)-3, E(X3)2, V(X) = 1, V(X:) = 5, V(Xs) = 2, and Y = 2X1 + X2-3X1 (a) Determine E(Y) and V(Y). P(Y > 2.0) and P( 1.3 Y 8.3).
3. [20 marks Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and Xs have the joint probability function (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks Find that the Fisher information matrix I(0). (c) [4 marks] Show that θ is an MVUE. (d) 4 marks Find the approximate distribution of Y 2X-X2, when the sample size n is large (e) [4 marks] Assume that X-(253, 234, 513). Find the...
11. Let Z = (X1,X2, X3)T be a portfolio of three assets. E(X) 0.50. E(X2-1.5. E(X3) = 2.5, VAR(X)-2, VAR(X2)-3, Var(Xs)-5·PX1.x2-0.6 and X1 and X2 are idependent of X3 (a) Find E(0.3xi +0.3X2 +0.4X3) and Var(0.3X1 +0.3X2 +0.4Xs) (b) Find P[0.3X1 +0.3x2 + 0.4X3 <2). Since z-table isn't provided, just write down the (c) Find the covariance between a portfolio that allocates 1/3 to each of the three assets and a portfolio that allocates 1/2 to each of the first...