PDA

View Full Version : Min/Max problem

ewr
September 1st 2005, 06:55 PM
I have a question on Actex Prob Set 4, #12. X1 and X2 are 2 indep. random variables, but they have the same density function
f(x) = 2x for 0<x<1, 0 elsewhere. Find the probability that the max of X1 and X2 is at least .5.

The solution says P[Max{X1,X2}>=.5]=1-P[(X1<.5) and (X2<.5)]

Why can't you just evaluate P[(X1>.5) and (X2>.5)]? I don't have a complete understanding of Min/Max problems and Order Statatisics, so if you have any kind of explanation, I'd appreciate it!

wat
September 1st 2005, 07:22 PM
I have a question on Actex Prob Set 4, #12. X1 and X2 are 2 indep. random variables, but they have the same density function
f(x) = 2x for 0<x<1, 0 elsewhere. Find the probability that the max of X1 and X2 is at least .5.

The solution says P[Max{X1,X2}>=.5]=1-P[(X1<.5) and (X2<.5)]

Why can't you just evaluate P[(X1>.5) and (X2>.5)]? I don't have a complete understanding of Min/Max problems and Order Statatisics, so if you have any kind of explanation, I'd appreciate it!

Your equation doesn't answer the question. Suppose for a minute that X1 = A and X2 = B (that way, we can use separate letters).

Your expression answers A and B both being greater than 0.5. The question is asking for the probability of at least one being greater than 0.5. If A was 0.7, but B was 0.3, is this something that you should count as a "success"? In other words, is Max{0.3, 0.7} >= 0.5? Yes. But for your expression, it wouldn't count as a success, since 0.7>=0.5, but not the case for 0.3.

Is that okay?

Sam Broverman
September 1st 2005, 07:23 PM
In order to have Max{X1,X2} > .5 , what we need is
either X1 > .5 or X2 > .5 , but we do not need both
to be true. The complement of the event Max{X1,X2} > .5
is the event that the max is <= .5 , so
P[Max{X1,X2}>.5] = 1 - P[Max{X1,X2}<=.5 ] .
In order for Max{X1,X2} <= .5 to be true, we must
have both X1 <= .5 and X2 <= .5 . This is an intersection
of the events (X1 <= .5) intersect (X2 <= .5) .
Because X1 and X2 are independent random variables,
the probability
P[(X1 <= .5) intersect (X2 <= .5)] can be written as
P(X1<=.5) x P(X2<=.5) .

ewr
September 1st 2005, 07:45 PM
Thank you for your responses, I appreciate your help!! The solution makes complete sense now!