00:01
So in this question we've got a sequence of independent random variables x n for n greater than or equal to 1, where x 2 n minus 1 are uniform on 0 to 1, and x 2 n are beta with parameters 2 and 1.
00:21
So that means that f x 2 n minus 1 x is 1 for x between 0 and 1 and 0 otherwise, and f x 2 n of x is 2t, or let's do these in terms of t, 2t for t between 0 and 1 and 0 otherwise.
01:04
So what's the probability of the event x 1 plus x 2 is greater than 3 over 2? well that's going to be the integral.
01:19
Now both of them can only be between 0 and 1, so we can integrate x 1 from 0 to 1, and x 2 has to be bigger than 3 halves minus x 1.
01:29
So 3 halves minus x 1 up to 1.
01:36
Now their densities are 1 and 2t, so we're going to have the product of those is ...
01:48
So for x 1 it's 1, and for x 2 it's 2t, so we'll do 2t and we integrate dt, and then we integrate dx 1.
02:01
So i'm integrating x 2 first, then x 1.
02:04
But actually, since x 1 is only between 0 and 1, 3 halves minus x 1 ...
02:13
Well it has to be smaller than 1, and that means that x 1 has to be bigger than 1 half.
02:20
So now we can integrate this.
02:33
2t integrates to t squared, so we have 1 minus 3 halves minus x 1 squared dx 1.
02:46
So expanding that out, we've ...
02:51
Well i can actually just integrate this term by term.
02:54
So integrating the first term, we get x 1.
02:59
Integrating the second term, we get plus 3 halves minus x 1 cubed times 1 third.
03:08
And that goes between 1 half and 1.
03:10
So for the first term, we get 1 minus 1 half, which is 1 half...