Pa Suppose that (X1, X2) ~ N(0,0,1,1,0). It follows from this that the joint PDF of...
Suppose that (X1, X2) N (0,0,1,1,0). It follows from this that the joint PDF of (X1, X2) is given by Ixvx:(21,2) = exp (1} (27 +23)) Furthermore, if 1 Y Tā (X1 + X2) and Y2 (X1 - X2) Then (Y1,Y) ~ N(0,0,1,1,0) as well. (a) If X1 <X2, what are the possible values of Y¡ and Y2? (b) If Y, <0, what are the possible values of Xi and X,? (c) What is the marginal distribution of Yg? (d)...
6. Suppose that (X1, X2) ~ N(0,0,1,1,0). It follows from this that the joint PDF of (X1, X2) is given by 1 fx1,x2 (x1, x2) - cxp («** + x2)) Furthermore, if 1 and (X1 + X2) ✓2 1 Y2 (X1 - X2) V2 Then (Yı, Y2) ~ N(0,0,1,1,0) as well. (a) If X1 < X2, what are the possible values of Y¡ and Y2? (b) If Y2 < 0, what are the possible values of X1 and X2? (c)...
12. (8 Pts.) Let Xi and X2 have the joint PDF Let Yi Xi/X2 and Y2 = Xy. Find the joint PDF of(H.)a). Are Y1 and Y2 independent?
Exercise 7 (team 5) Let Xi and X2 have joint pdf x1 + x2 if0<x1 < 1 and 0 < x2 < 1 /h.x2 (x1,x2) = 0 otherwise. When Y1 X1X2 derive the marginal pdf for Y.
Let X1 and X2 have a joint pdf
Let
Find the joint pdf of Y1 and Y2.
f(x, y) = + y, 0<x,y<1
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...
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... < Yn be the order statistics from this sample. a) Find the joint pdf of (Y1, .. , Yn). b) Find the joint pdf of (W1, .. , Wn) where W1 = nY1, W2 = (n-1)(Y2 -Y1), W3 = (n - 2)(Y3 - Y2),..., Wn-1 = 2(Yn-1 - Yn-2), Wn = Yn - Yn-1. (c) Show that Wi's are independent and its distribution is identically...
Let Xi, X2, X3 be i.id. N(0.1) Suppose Yı = Xi + X2 + X3,Ý, = Xi-X2, у,-X,-X3. Find the joint pdf of Y-(y, Ya, y), using: andom variables. a. The method of variable transformations (Jacobian), b. Multivariate normal distribution properties.