Question

Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that P(X=a)=p1, P(max(X, Y) =a) =p2, and P(min(X, Y) =a) =p3.

Find P(Y=a) in terms of p1, p2 and p3.

Hint: Use the first Bayes theorem. Choose a suitable event-complement pair to partition the probability space by comparing X and Y .

Answer 1

Complete solution for the problem can be found in the images below with all the necessary steps.
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こ P (Y= a) P(Y=a/xsy). PCX5Y) +P [ Y=0 / x70Y) -P(x>Y) P(Y Man(X,Y)= ) P( mon(x,y)= a) = P(x=a ,Y<a) + P(Y=a, x < a) +1(x=a, Y=0 ☺ Y=a). 6. & P ( min (X,Y)= a P(x=9, Y>a) + P (Y=a, Xa) + P(x=9, Y=a)

(1L we Adding 0 s get, Р{~~[9, 1)- а) + (-m(x,)) (а, ч<b) + ( Yen, xce ) +P (у-а,154) +Р (Y-а , X}a +24 Р Х = 4, Y = 4 а). х

7 P(x=a, Y>a) + P(x=a Y=a) + P(x=a, y <a pl L (un Noe Z IV . > PLY= a) P (Y= a, xya) + P (Y=a. X<a a) + P ( 4 = a, X=a) from 9 8 Thias, P(x=a + PLY=a) =) p1 + P(za) = {p(x29,779 + Pt x=9, YLa + folyza , X7a) + f (tza, Xtaj + PLY=a, X29) ::b1 + P(Y=a) = (mon an(X, Y) = 1 a) + P(min. (x, y) = a L Cfrom ] Y a) + P(x=9, Y =9)

7. PC PC 7- а Ya ) = P2 + F3 21.