00:01
For this problem, we have a probability distribution function given by a piecewise definition.
00:09
So we have x over 2, 1 over 2, 3 minus x over 2, and 0.
00:16
We have x over 2 for x between 0 and 1, 1 over 2 for x between 1 and 2, 3 minus x over 2 for x between 2 and 3, and 0 everywhere else.
00:37
In order to get our distribution function, the cumulative distribution function, what we'll have to do is integrate over each one of those regions.
00:46
So to begin, we want to take the integral from 0 to x of t over 2 dt.
00:57
Introducing t as our dummy variable here.
01:00
So that would become t squared over 4, evaluated from 0 to x, or simply x squared over 4.
01:10
We note that the total value at the end point of that initial region, total so far, is going to be 1 squared over 4, so the total is just going to be 1 over 4.
01:23
Then, for the next region, we'll need to integrate our probability distribution function there, plus, or then we add on the existing total.
01:37
So we have that this is just going to be t over 2 plus, or pardon me, i need to be careful in specifying my boundaries here.
01:48
So this is going to be t over 2, evaluated from 1 to 2, plus 1 over 4, that's our cumulative probability in this region.
01:57
So that would become, or actually that should be evaluated from 1 to x.
02:01
So that's going to be x over 2 minus 1 over 2 plus 1 over 4, so that's going to turn to x over 2 minus 1 over 4...