Solution :
E(Y) = 5 - E(X/3) - E(X)
= 5 - 1/3 * E(X) - E(X)
= 5 - 1/3 * 5 - 5
= -5 / 3
Mean of Y is -5/3 = -1.67
If E(X) 5, V(X) = 3, and Y XX, what is the variance of Y? 5
Suppose X and Y V(X) = 3 and V(Y ) = 5. Find: (a) V(X + Y ) given that X, Y are independent (b) V(3X + 4) given that X, Y are independent (c) V(X + X) given that E(X · Y ) = −1 (d) V(X + 3Y ) given that E(X · Y ) = 0
Suppose a discrete random variable y has the mean E(Y)=5 and V(Y)=49. What is the value of V(-2Y+3)?
1. Suppose that E(X) E(Y) E(Z) 2 Y and Z are independent, Cov(X, Y) V(X) V(Z) 4, V(Y) = 3 Let U X 3Y +Z and W = 2X + Y + Z 1, and Cov(X, Z) = -1 Compute E(U) and V (U) b. Compute Cov(U, W). а.
8. Let V = {(x,y)x,y e R}. Define addition on V as follows: (x,x)+(x2,)=(x, +x,-1,, +y,+3) [4 marks] a. Prove addition axiom #3 (Addition is commutative). b. Find the zero vector.
5. Let V = {(x + 2y, x + 2y) : x,y,z E R} be a subspace of R2, Find dim V.
Let XN(0, 1) and Y eX. (X) (a) Find E[Y] and V(Y). (b) Compute the approximate values of E[Y] and V(Y) using E(X)(u)+"()VX) and V((X))b(u)2V(X). Do you expect good approximations? Justify your an- Swer Let XN(0, 1) and Y eX. (X) (a) Find E[Y] and V(Y). (b) Compute the approximate values of E[Y] and V(Y) using E(X)(u)+"()VX) and V((X))b(u)2V(X). Do you expect good approximations? Justify your an- Swer
Suppose (X, Y ) has bivariate normal distribution, E(X) = E(Y ) = 0,V ar(X) = σX2 , V ar(Y ) = σY2 and Correl(X, Y ) = ρ. Calculate the conditional expectation E(X2|Y ). I. Suppose (X,Y) has bivariate normal distribution, E(X) = E(Y) 0, Var(X)-σ , Var(Y) σ and Correl (X,Y)-p. Calculate the conditional expectation ECKY expectation E(X2Y)
tion? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(a) P(X s a) for a (0, 1]. (4) Let Y =-log X. Calculate F(y)-P(Y v) for u 20. Calculate the density of Y. tion? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(a) P(X s a) for a (0, 1]. (4) Let Y =-log X. Calculate F(y)-P(Y v) for u 20. Calculate the density of Y.
E = "Expected Value" V = "Variance" 0 < x < 00, x < y < oo IS joint probability density function a) Compute the probability that X < 1 and Y < 2. b) Find E(X) c) Find E(Y d) Find V(X) e) Find V(Y)