Order Statistics, N independent uniform random variables

Hi everyone,

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:

0.8^N

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?

Thanks

Addendum to my proposed solution

P(M > 0.7 (intersection) (X1,...,XN) <= 0.8)/ (0.8^N)

I think I have a proposed way to get the intersection.

Since the maximum will be greater 0.7 and we know all the variables have an upper limit of 0.8.

The cumulative distribution function of the maximum will be evaluated with the limits of integration being 0.7 for the lower bound and 0.8 for the upper bound.

The pdf of the maximum of N independent uniform random variables is given by

f(x) = n * (x^(n-1))

So the integral from 0.7 to 0.8 will give us .

And taking the denominator I mentioned above getting a final answer of

Does this line of reasoning suit everyone?

My proposed solution doesn't seem right

I don't think my solution is correct because its implying as N -> infinity, the probability approaches 1. That doesn't seem correct. Any thoughts?