FRI (2+2+3=7 points) Suppose (X1,X2) has joint pmf 9! za2(9-21-1/2)9-71-+9, X1,X2(T1, 2) where z1...
Let X1, X2, X3 be independent Binomial(3,p) random variables. Define Y1 = X1 + X3 and Y2 = X2 + X3. Define Z1 = 1 if Y1 = 0; and 0 otherwise. Define Z2 = 1 if Y2 = 0; and 0 otherwise. As Z1 and Z3 both contain X3, are Z1 and Z3 independent? What is the marginal PMF of Z1 and Z2 and joint PMF of (Z1, Z2) and what is the correlation coefficient between Z1 and Z2?
2. Let X1 and X2 have the joint pmf p(11, 12) = 232, 21 = 1,2,3 and x2 = 1,2,3, zero elsewhere. Find the joint pmf of Y1 = X1 X2 and Y2 = X2, and then find the marginal pmf of Y.
= = 3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, Let Var(X1) = Var(X3) = 2, Var(X2) Cov(X1, X3) = -1. i) Suppose Y1 = X1 - X2. Find Var(Y1). ii) Suppose Y2 = X1 – 2X2 – X3. Find Var(Y2) and Cov(Yı, Y2). Assuming that (X1, X2, X3) are multivariate normal, with mean 0 and covariances as specified above, find the joint density function fxı,Y,(y1, y2). iii) Suppose Y3 = X1 + X2 + X3. Compute the covariance...
1. Suppose that X1, X2, and X3 E(X1) = 0, E(X2) = 1, E(X3) = 1, Var(X1) = 1, Var(X2) = 2, Var(X3) = 3, Cov(X1, X2) = -1, Cov(X2, X3) = 1, where X1 and X3 are independent. a.) Find the covariance cov(X1 + X2, X1 - X3). b.) Define U = 2X1 - X2 + X3. Find the mean and variance of U.
MULTIVARIATE DISTRIBUTIONS
3. Suppose that Xi and X2 are independent and each has a uniform distribution on (0,1). Define Y: X1 + X2 and Y2 = X1-X2. Find the marginal probability density functions of Y1 and Y2. . Suppose that X has a standard normal distribution, and that the conditional distribution of Y given X is a normal distribution with mean 2X 3 and variance 12. Find E(Y) and Var(Y)
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3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, 5. Let Var(x1) = Var(X3) = 2, Var(X2) Cov(X1, X3) = -1. i) Suppose Y1 = X1 - X2. Find Var(Y1). ii) Suppose Y2 = X1 – 2X2 – X3. Find Var(Y2) and Cov(Y1, Y2). Assuming that (X1, X2, X3) are multivariate normal, with mean 0 and covariances as specified above, find the joint density function fyy, y,(91, y2). iii) Suppose Y3 =...
16. Find the direction of the force between Q1-5uC r1 (x1-2,y1-3,z1-3) and Q2-4uC r2 (x2-2, y2-3,z2-2) A) 0i 0j-1.3k 21 B) OiOj-56k R12 0 c) 0i 0j-1.45k 17. Find the force (vector) between Q1-33uCr1 (x1-1, y1-2, z1-3) and Q2-63uC r2 (x2-3, y2-3,z2-1) A) .76i .38j -.77k F2 B) .971 48j-.98k R12 C) 1.17i .58j-1.18k D) 1.38i .69-1.39k
16. Find the direction of the force between Q1-5uC r1 (x1-2,y1-3,z1-3) and Q2-4uC r2 (x2-2, y2-3,z2-2) A) 0i 0j-1.3k 21 B) OiOj-56k R12...
Suppose X1 and X2 are continuous random variables with X1 ~ Unif(0, 1), X2 | X1 = x1 ~ Unif(0, X1) (a) Find the pdf for the joint distribution of X1 and X2 (b) Find the pdf for the marginal distribution of X1 (c) Find the pdf for the marginal distribution of X2 (d) Find the pdf for the conditional distribution of X1 | X2 = x2 (e) Write 1 or 2 sentences explaining how this problem relates to Bayes’...
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).
6. Suppose random variables X1, X2, X3 have the following properties: E(X1) = 1; E(X2) = 2; E(X3) = −1 V(X1) = 1; V(X2) = 3; V(X3) = 5 COV (X1,X2) = 7; COV (X1,X3) = −4; COV (X2,X3) = 2 Let U = X1 −2X2 + X3 and W = 3X1 + X2. (a) Find V(U) (b) Find COV (U,W).