Question

Escalate - Show every mynute steps in DETAILED! Explain & SHOW how the LIMITS of integration & DOMAIN are found.

Let X1, X2, and X3 be independent Uniform(0,1)-distributed random variables. (a) Find the joint pdf of (X (1), X(3)). Remark: Pay attention to the domain of the joint pdf. (b) Find the conditional pdf of X(3) given that X (1) = 1/2. Remark: Pay attention to the domain of the joint pdf. (e) Find P(X(3) > 2/3 |X(1) = 1/2). Remark: Pay attention to the integration domain.

Answer 1

20/ Page 1 - X1, X2, X3, X 4 lid and follow Ulo,1) F(x) = S = 1 jola To, ow Now (DF gf Unyom (0,1) F(x) = CS*F(x)dx = [ ide Now we arrange given Variable in asunding order Xen ? X12) 2X63) ? X14) omula. Now we know for Order stadiubid ✓ X1, X2... Yn üd e arrange in amending order I Xco? X(2)... LX(K)? . Xen) pdf of X(K) – In [P(X(X)]* [P(xxx) f(x) | |к- Ink in [F60[1-FC) for) ha TK-1 Int polf oy X(1) = n [1-F007" for) - 4X (1-23 pdf of X(3) = 2 x² (1-x) = 12 x (1 x) L2 L

on Now we also know & Dra? Page 2 We also know for loint pdf of Ordered sladuties (1) X1 < X (2) L... {X(K)? ... {XoK. & Xm) Joint polt of (x,y) = ln [Foo]* {P/2(XLG) C - Fly) Falfly) [K-1 |x-kinor pdf of (xan, Y) = 4 redi] [-2] Flxan) Flyes) - 2 x 3 (4-X) (1-x) x 2² x3 441-4) x2² (1-70 = 2 x 9 (4-x) (1-2) (1-x) e OLX2X13) (1) IF (X, Y (b) Conditional pdf g X29) given Yes z . = F(X13) X(1) = F(X13/%) = F(X) X(31) - 2 kg g (2 - x)(-y) (1-x) & F(x)) - 4*(1-x) - 2 x 9 y ly x)(1-4) (1-x) IF (X2)X = 8) z 22 9 y 12 - 14) (1-4) * ! (1948*(1-2)(x-1)) Bye

(270-27-090]člo BP ( By Ro - Ag], Jazz TAPO =Pz-Ke), JOLE Kp (- RE) (5-125 L, J = hp Rheon 3.600 J = ( 3 = " \ && (@xld (7)
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