Suppose that (X1, X2) N (0,0,1,1,0). It follows from this that the joint PDF of (X1,...
Pa Suppose that (X1, X2) ~ N(0,0,1,1,0). It follows from this that the joint PDF of (X1, Xy) is given by fxıx(31,09) - exp(+34 +23) Furthermore, if GE Thte and 12(x3 + x2) - tao V5 (Xi-X) que Then (Y1,Y,) ~ N(0,0, 1, 1,0) as well. (a) If X, < X2, what are the possible values of Y1 and Y2? (b) If Y2 < 0, what are the possible values of X, and X,? (c) What is the marginal distribution...
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)...
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
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
X1 and X2 are IID with the density: a) joint density with r.v. Y1 = X1 and Y2 = X1 + X2 (I think this might be transformation but I'm stuck after) b) marginal density for Y2 f(x) = V2nz exp (-2) , for x>0,
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...
(a) Write down the joint pdf of X1 and X2. [4] (b) By using the transformation of random variable method, find the joint pdf of Y1 = X1 and Y2 = X2/X1. [16] (c) Hence find the marginal pdfs of Y1 and Y2. [8] (d) Compute the covariance between Y1 and Y2, cov [Y1, Y2]. [8] (e) State, with justification, whether Y1 and Y2 are independent.
4. Suppose that X and X2 have joint PDF: jXiXy(Xi,T2)=Í : otherwise 0 (a) Use the transformation technique to find the joint PDF of Y1 and Y2 where Yi = (b) Using your answer to part (a), find and identify the distribution of Yi
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.