00:01
For this problem, we can use the method of transformation of random variables to find the probability density function of y.
00:14
So the thing that we'll do here is we first want to find the cumulative distribution function, the cdf for y.
00:22
We have that the cumulative distribution function is going to be the probability of capital y less than or equal to little y, where we know that capital y, the actual value of the random variable, is going to be the same thing as e to the power of negative x.
00:39
So we're looking for a probability of e to the power of negative x less than or equal to little y.
00:46
We can rearrange this to get an expression for x in terms of y by taking the logarithm of both sides.
00:57
We'll get negative e to the power of, oops, not negative e to the power of, we'll get negative x times ln of e, which is just 1, must be less than or equal to ln of y.
01:19
If we divide both sides by negative 1 to swap around, or to isolate x, then that will also swap around the inequality.
01:29
So we get that x must be greater than or equal to negative ln of y.
01:35
Probability of x greater than or equal to negative ln of y, we can find by taking 1 minus the cumulative distribution function of x, evaluated at ln of y, or at negative ln of y rather.
01:50
And so we now have part of our expression here.
01:55
To find the cumulative distribution function, well we need to keep in mind that x is uniformly distributed from 0 to 1.
02:05
So that means that the cumulative distribution function for x is just going to be, well well, it actually just works out to being equal to x...