TOPIC: Use of mean vector and the dispersion matrix of a random vector to find the values required.
Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 3...
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)
how to calculate cov(x1,x2), cov(x2,x3),cov(x3,x1)?
and how to calculate var(x1),var(x2),var(x3)?
Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 E [X] ,1-10 | and var(X)=Σ-| 0 3 0. 1 0.5 1 compuite: 2
2 0 -1 EIX) = μ = | 0 | and var(X) Σ _ | 0 -1 0.5 3 0.5 | compute: (a) E[Xi +Xs] (c) var(X2- X3) d var(X2 + X) (e) cov(4X2 +X1,3Xi -2X)
2 0 -1 EIX) = μ = | 0 | and var(X) Σ _ | 0 -1 0.5 3 0.5 | compute: (a) E[Xi +Xs] (c) var(X2- X3) d var(X2 + X) (e) cov(4X2 +X1,3Xi -2X)
1 [3]. Let X1,X2, X3 be iid random variables with the common mean --1 2-4 and variance σ Find (a) E (2X1 - 3X2 + 4X3); (b) Var(2X1 -4X2); (c) Cov(Xi - X2, X1 +2X2).
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...
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...
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,...
3. Let X1, X2, . . . , Xn be random variables with a common mean μ. Sup- pose that cov[Xi, xj] = 0 for all i and A such that j > i+1. If 仁1 and 6 VECTORS OF RANDOM VARIABLES prove that = var X n(n- 3)