00:01
In this question, we have that xn are exponential random variables with a mean of n, so with a parameter 1 over n, and this is valid for n greater than or equal to 1.
00:17
And then we have fn, which is the distribution function of a variable.
00:27
So it's the distribution of xn minus 10 times the floor function of xn over 10.
00:39
Now what does this mean? well, let's call this yn.
00:45
Well, what is this part doing here? what we're doing is, if we say that we can write xn is 10s plus t, where s is an integer from 1 to 10, sorry, from 0 to 9, well s is just some integer and t is an integer from 0 to 9, then when we take xn over 10 and we take the floor function, then this is equal to s.
01:34
So yn is actually going to be 10s plus t minus 10s, because dividing xn by 10, we're going to get s plus t over 10, but t over 10 is between 0 and 9.
01:53
So that's going to go away when we take the floor function.
01:58
Then multiplying by 10 again, we just get our 10s.
02:01
We take that off and we get the t.
02:08
Actually, t doesn't have to be an integer.
02:11
T is just a real number from 0 to 10, but not including 10.
02:21
So yn is the remainder when we divide xn by 10.
02:25
So what's the probability, what's the distribution function for yn? well, fn of y is the probability that yn is less than or equal to y, which is the sum from i equals 0 to infinity, the probability that xn is between 10i and 10i plus y.
03:02
So this is going to be the sum from i equals 0 to infinity of the integral from 10i to 10i plus y of the density function for the xn's, which is 1 over n e to the minus x over n dx.
03:25
So now we can do that integral...