Thread: Probability question

Hey downthesun01.

You will have to look at two situations: the first is when x is in [0,1] and the second is when x > 1. You will have two situations and you need to find the maximum value of both situations separately.

In the second situation F1 will be a constant (i.e. 1).

Thanks. I see what you're saying about the two situations. How do I go about finding the maximum value of each situation? Am I taking the derivative of each situation?

Expand out the absolute value function so that it's analytic (i.e. |x| = x if x > 0 and -x if x < 0) and then use standard calculus to find the maximum point.

Hmm.. For the situation involving [0,1], I've tried, but I'm getting x= 0, -1 while the book gives x=0, 1

The solution in the manual lacks any real intermediate steps, I'm left kind of stumped right now

You can't get -1 since the range is in [0,infinity). Can you show us how you got -1 as a solution?

I can't find where I wrote everything down at the moment, but since the range is shouldn't absolute value function be

?

After that, I thought that I take the derivative, which is



Set it equal to 0, and find the values x=c such that

You can only use that function in the interval [0,1]. In the interval (1,infinity) you will use |1 - 1/(1+x) - 1| = |1/(x+1)| = dist(F1,F2) for x > 1.

Sorry, I meant [0,1]

The manual's solution says that it's maximized at either , or a critical point; critical points occur where or .

So, I guess:



I don't see how they got 1 though

That takes care of x = 0.

For x = 1 case we look at maximizing |1 - 1/(x+1)^2 - 1| or |1/(x+1)^2|

If x > 1 then |1/(x+1)^2| = 1/(x+1)^2.

We can show that the function is decreasing for all x > -1 since d/dx 1/(x+1)^2 = -2/(x+1)^3 < 0 for x >= 1.

This implies that the maximum value for x >= 1 is at x = 1 and thus we have the second solution of x = 1.