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1. For pdf f (r, y) = 1.22, 0 < x < 1,0 < y < 2, z +y > 1, calculate: EY) and () E (X2)
Let X. Y be two random variables with joint density fx.x(x,y) = 2(x + y), 0<x<y<1 = 0, OTHERWISE a) Find the density of Z = X-Y b) Find the conditional density of fXlY (x|y) c)Find E[X|Y (x|y)] d) Calculate Cov(X, Z)
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
Let X and Y have join density
6 f(x, y) =-(x + y)2, 0 < x < 1, 0 < y < 1
20 -{24R/1<x<1+ }}-(1,1+ 4) for each positive integer S: = ? a. US = ? i=1 4 b. O S = ? 11 c. Are S1, S2, S3, ... mutually disjoint? Explain. d. ÜS; = ? =1 72 e. n S = ? 1-1 00 f. US = ? 00 g. S; = ?
Suppose S CR3 is the intersection of B(0; 2) and the cylinder {(1, y, z) : y2 + 22 <1}, and that 1 the density of S is given by p(x, y, z) = -2 5. Set up an iterated integral which gives the mass of S (you do not need to evaluate it).
(10 pts) The joint distribution of X and Y is given by: f(x,y) = 1/y, 0 < x < y < 1. Derive the distribution of Z= Y/X. You must use both the methods (CDF & Transforma- tion).
sint, 0<t〈π . У(0)=1, y'(0)=0
( xy 7. CHALLENGE: fxy(x, y) = 0< < 2, 0 <y <1 otherwise 0 Find P(X+Y < 1) HINT: consider the region of the XY plane where the inequality is true.
S 6 Calculate the fleux SSF. ds, where I la, y, z)= <0? 2², 227 and S is the finite cylinder (with top and bottom) given by x² + y² = 1, 2 = 0, Z = 3,