00:01
So in this question we're given a probability distribution for a random variable, which is this.
00:09
And first, and we're told that we have another random variable, y equals 2x plus 10.
00:16
Now what we know is that the probability that x is in a region from x to x plus dx is going to be f of x evaluated at x times d x.
00:34
Now, the probability that y is inside a region from y to y plus d y is also going to be f of y evaluated at that value y times d y.
00:49
But y is 2x plus 10.
00:52
So this probability is that x is in a region between y minus 10 divided by 2.
01:03
So y over 2 minus 5 to y over 2 minus 5 plus d .y over 2 minus 5.
01:20
Sorry, y plus dy over 2 minus 5.
01:22
So plus dy over 2.
01:26
And this is f of y, dy, dy.
01:30
But we also know that this is fx of y over 2 minus 5 times dy over 2.
01:40
This is f x y over 2 minus 5, dy over 2.
01:48
So what that tells you is that f y of y is f x of y over 2 minus 5 divided by 2.
02:02
So that means that f y of y is going to be y over 2 minus 5 divided by 18 divided by 2 whenever y over 2 minus 5 is between 0 and 6.
02:24
So this is, so multiplying the top and the bottom by 2, we get y minus 10 divided by 4 times 18.
02:35
Actually let's expand that out.
02:38
So we're going to get y over 4 times 18.
02:40
So 4 times 18 is 72.
02:46
And then here we get minus 5 over 2 times 18.
02:49
So 5 over 2 times 18.
02:50
So 5 over 2 times 18...