00:01
Alright, in your question, you're given the probability density function for some random variable x, and it's 4x cubed from when x is between 0 and 1, and it's 0 otherwise.
00:12
So what we're really focused on here is the 4x cubed function, and when we find the area from 0 to 1 under this curve, it actually equals 1, which means it counts for all possible probabilities of that random variable.
00:36
Now what you're focused on is trying to say, what is the probability that it's less than two thirds if we know it's greater than one third? so we want, this would be our possible area that we know we're in, and then we would want to figure out, i'll change color here, what fraction of that area is, contained in here in red.
01:07
I'll show that in symbols in a little bit, but we need to be able to calculate the area under that curve.
01:14
To do that, we're going to use an integral from a to b of the function.
01:22
We'll take the antiderivative, which would be 4x to the 4th, divided by 4.
01:28
That cancels out the 4, giving us x to the 4th, from a to b, and that represents the area under this curve.
01:44
Now, your probability, it's a conditional probability.
01:49
So what would be how we would set this up is the probability of x being greater than one third and less than or equal to two thirds.
02:02
And we divide it by the probability of it being greater than one third alone.
02:10
So for the numerator that's just going to be x to the fourth from one third to two thirds.
02:18
For the denominator that's going to be x to the fourth from one third to two thirds...