All right.
It is example time.
Let's look at the scenario here.
We have X big acts, of course, being our continuous random variable, and we're told that it's PDF.
The probability density function is equal to three x squared and the interval we're interested in the
values that we're interested in is from zero to one, including zero and one.
So the question is what is the probability of the X being between half and one?
So let's go back to that handy notation and understand what we're looking at.
We want the probability that the ax is between half and one, right?
And then.
We have our integration from where from half to one.
And then we have the PDF, which is three x squared with respect to X.
So now this is the tricky part and the part that you haven't had a chance to do so far, but it is very
simple in this case where we have to do is we have to integrate when we integrate threats to what we
do is we increase the index here.
So it's squared, we increase the index by one, so it becomes three x to the three and then we divide
by that increased number three.
So this cancels out and we're left with x cubed.
So I'll just get rid of that and leave us with what we are left with when we integrate, so we have
access to the three.
And we put that in these brackets.
And then on the side, we just put half to one.
And then we are just doing the calculation, so we have one cubed minus half cubed.
And that equals seven eighths.
Now, whenever you get to the answer and you compute your answer, you want to always think about it
and does it pass the smell test?
The best way to do that is to sketch out the function here three x squared and see if it makes any sense.
Does seven eight seven eighths makes sense in this case for the probability of X between half and one
to be seven eighths?
Almost all of it?
Well, let's see.
So if you sketch three x squared, you get this parabola.
The thing is our domain.
So our interval in terms of the X values that were interested in the range is between zero and one.
So what that means is that we're only interested in from here to here.
OK.
We are not interested.
In the negative X values here, none of this, we're only interested in this range and in particular
what we're interested in.
I'll try to make a straight line down, but it's can be hard.
There we go.
We're only interested in where one point five is right around here.
So.
Does this look to you like this is seven eighths of the total?
She certainly does, because here's the other part the other part is right here.
So this is one eighth.
Makes perfect sense when you actually sketch it out and see what you're dealing with.