I apologize as I don't know LaTex, so formatting will be off, I'll try my best though.
Here is the question:
Let X1, ...,Xn be independent uniform random variables in (0, 1). Also, let
M = max(X1, ...,Xn) and L = min(X1, ...,Xn).
Find the probability that the maximum is greater than 0.7 if you already
know that all X1, ...,Xn are less or equal to 0.8
My attempt at a solution:
P(M > 0.7 | (X1,...,XN) <= 0.8)
= P(M > 0.7 (intersection) (X1,...,XN) <= 0.8)/ (P((X1,...,XN) <= 0.8))
I know the denominator is equivalent to saying:
P(X1 <=0.8) and P(X2 <=0.8)....P(XN <=0.8)
Since they are independent the cumulative probabilities can be multiplied to obtain to obtain the value of the denominator.
So take the integral from 0 to 0.8 of a uniform distribution, which will just give you 0.8. Since there are N urv's we get:
Therefore we have
P(M > 0.7 (intersection) (X1,...,XN) <= 0.8)/ (0.8^N)
Now the part I'm having trouble with is the set of intersection, can we simply multiply the two probabilities in the numerator to obtain the intersection probability? I don't think we can, I just can't pin it down properly. Any insights?