4 0 05 0 -10 2 2. If X - N3 (μ, E) where μ = [1-1 2] and Σ = Which of the following random vectors are independent? (i) X₁ and X₂ (ii) X₁ and X3 (iii) X2 and X3 (iv) X₁ X3 and X2
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![2. If XN3 (μ, E) where u¹ = [1 -1 2] and Σ = [
40 -1]
0 5 0
-1
0
2
Which of the following random vectors are independent?
(i) X₁ and X₂ (ii) X₁ and X3 (iii) X2 and X3 (iv) X₁ X3 and X2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c8b177e-084a-4634-9609-77aef3900f8a%2Fd679e008-3ded-424e-a383-f71814440f38%2Fnmi66_processed.jpeg&w=3840&q=75)
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- Let X1, X2, X3 be random variables each having a mean u and variance o. Further, Cov(X1, X2) = 2, Cov(X1.X3) = 3 and Cov(X2, X3) = 1 Define U = 2X1 – X2 + 4 X3 Solve for the mean and standard deviation of U.Suppose that X and Y are random variables with E(X) 2 , E(Y) = 5 and E(x?) = 8, E(Y?) = 30 and cov(2X – Y,4X – 3Y) = -12, then cov(-5X – 3,2Y + 4) is equal to: Hint: cov(ax + bY,cX + dY) = acV(X) + bdV(Y) + (bc + ad)cov(X,Y) O -120 O -59 O -64 O -302. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
- c. Suppose that the random variables X1, X2 and X3 are independent with means and variances as follows: H1 = 8 of = 36 H2 = 5 o = 81 H3 = 7 o3 = 49 Let Y1 = X1 + 2X, and Y2 = 2X1 – }X2 – X3. (i) Find the Cov(Y1, Y2). (ii) Assuming that X1, X2 and X3 are Normal random variables, calculate P(Y < 2Y,).TRUE OR FALSE. a.) Let X (X₁, X2,..., Xn)' be a random vector with joint cumulative distribution function 2 - fx(-). Then (Xi − #i)² has a chi-square distribution with n degrees of freedom where 72 i=1 Hand o are the mean and variance, respectively, of each of the random variable in X, i = 1,2,..., n. b.) Let X₁, X2,..., Xn be a random sample from a density f(), which has mean and finite n 0² n variance o2, and let X = n X₁. Then E[X] = µ and Var(X) = μ = c.) For any random variables X and Y, E[XY] = E[X] E[Y].ii) Let X be a random variable taking three values: P(X= a₁) = P₁, P(X=a₂) = P2, P(X=a3) = P3, where p₁ + P2 + P3 = 1 and P₁, P2, P3 € (0, 1). Let A = {X = a₁} and G = {N, 0, A, Ac}. Prove that E (X³|G) = a³¹₁ + ª²P² + a²P3, P2 + P3 1A.
- Suppose that X and Y are random variables with E(X) = 2, E(Y) = 5 and E(X) = 8, E(Y) = 30 and cov(X + 2Y,-3X+ 4Y) = -12, then cov(2X +3,-3Y +4) is equal to: Hint: cov(aX + bY.cX + dY) = acV(X) + bdV(Y) + (bc + ad)cov(X, Y) -59 O -120 -64 -303. Let the random variable X have the pmf f(x) = a) E(X) b) E(X²) c) E(3X²2X + 4) (x+1)² 9 for x = -1,0,1. ComputeSuppose that X and Y are random variables with E(X) = 2 , E(Y) = 5 and E(X²) = 8, E(Y?) = 30 and cov(X – 2Y,3X – 4Y) = -12, then cov(2X – 3, –5Y + 4) is equal to: Hint: cov(aX + bY,cx + dY) = acV(X) + bdV(Y) + (bc + ad)cov(X,Y) -59 -120 -64 -30
- Let X1, X2, X3 be random variables such that V ar(X1) = 5, V ar(X2) = 4,V ar(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = −2 and X2 and X3 are independent. Findthe covariance between Y1 = X1 − 2X2 + 3X3 and Y2 = −2X1 + 3X2 + 4X3Let x1 and x1 be independent standard normal random variables. If Y1= x1 + 2x2 and Y2 = 2x1 - x2, find Fy1,y2(1,2)Let X1, X2, X3 be random variables such that Var(X1) = 5, Var(X2) = 4, Var(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = -2 and X2 and X3 are independent. Find the covariance between Y1 = X1 – 2X2 + 3X3 and Y2 = -2X1 + 3X2 + 4X3. %3D
