Question

Exercise Consider the sample space S = {(x,y) ERP: + y<1}, with event space & suitably chosen, and with probability measure P determined by Area(E) Area(E) P(E) Area(S) for E E E. Let X: S+R be the random variable defined by (-1/2 if : <0 and y 0, if : <0 and y <0, X((x,y)) = if x > 0 and y < 0, V2? + y2 if x 20 and y 2 0. Find the probability mass and density functions associated to X and use them to compute the following statistics: (i) E[X] (ii) Var[X] (iii) E(X) (iv) E (2x + 3) 1

Topic: Expected value, variance, and moment generating functions.

Answer 1

ANSWER:

0 B A C D here s is divided in down to area each have equal area. ㅠ002 FIT 니 P[B] -. P[xcoo] TI Area of s I. 니 ㅠ PCB) =) Y Similarly, Pcc) - P.[yco ,u<0] = ㅒ에 다 4 ㅠ PCD) = P(x>0, co] - . ET Y P(A) = Pfx20, U20) 니 vàndom variable x. (014) and its probability dh is given by

2 X(214) -112 P(X) V3 2 Yu. 1 4 √x² + y2 14 ELX)= ( x if cons it Ženy 69x5 2X 12 + SS Joy? dady x20 420 clearly dady into polar O = 1/2 x=rcoso y - Ysino -x² + y2 = 82 0-0 7/21 :i)EX). - 8 + S Fordrdo ។ 0-0 2-0 T/2 3+2+12 t (9) de 24. it X 2 4 24 TT llt 24. 16 31 +22 48 Ex)= 0.65 o

Elx?) ($*+63 )** +(2)2x} + SS(2492) & dedy T12 ! + 82 ч dr do V . tit 36 16 712 + 9+4+144 4x4 xЧ Sin de o T 157 at 32 144 Elx?) 1.188 312 (x442) dx dy *) ELX3): - a to vý +-8 ***S$ S 4 wo yoo 3/2 1 - 27 it'st s 8317 dr do 8 O=0 ro ${** *****] TEX)= 2.05 [{(2x+373) · EC 8x3+27 + 3 (235 - 37 3123), 32) 3 iv)

(u *FC832+27+ 3632154 ) ESÊ (23)+27 +3 6E (52) +54 ECX) from 0,43 8x(9.05 ) +27 +36(1.188 ) +54x10.65) @[12343)– 12.1.2-68 1) V(X) = E(X2)-(ECX)].? 31.188 - 10165)? V(x) = 0 20.7655