00:01
So we're given the following probability or cumulative distribution function for a random variable x.
00:08
And we need to figure out c such as this becomes a valid cumulative distribution function.
00:14
And so with a cumulative distribution function, it has to start with zero and end at one to be valid.
00:25
And it has to always be positive, right? and so here we see that the function.
00:33
Evaluate to 0 when x is less than or equal to 0, and the function evaluates to x squared when x is between 0 and 1.
00:41
That means it's continuous such that f of 0 is equal to 0, which is told here, but then the limit as x approaches 0 from the right with x squared is also 0.
01:03
I'm sorry, so 0 squared equals 0, and then as the limit of this part as x approach is 1, f of 1 is equal to 1 squared, which is 1.
01:15
And we want this to be continuous, and we also want it to continue at 1 indefinitely.
01:20
So the value of c must equal 1.
01:24
So now let's derive our probability density function from the cumulative distribution function.
01:32
And so the difference between the cumulative distribution function, which looks at the something like this, where this is zero, this is one, and this is one.
01:47
This is describing, cumulatively speaking, what is the probability of a certain value of x, up until x equals 1, where the probability that x is lessen or equal to 1 is 100, and the probability that x is less than or equal to anything greater than 1 is also 1 or 100%.
02:08
So this is the area under the probability density curve, which might look something more like this.
02:21
It would be piecewise, actually, sorry.
02:25
It would be something more like this.
02:27
So the probability density function is the derivative of the cumulative distribution function.
02:36
And so this is also done piecewise.
02:39
So f of x equals the derivative, the density function, which is 0 when x is less than or equal to 0 because the derivative of 0 is 0.
02:55
And then it's 2x from 0 is less than x is less than x is less than or equal to 1.
03:01
And it's 0 again when x is greater than 1.
03:06
And for this to be a valid probability density function, it has to always be greater than or equal to 0, which it is.
03:15
And the area under the curve must equal 1.
03:18
So if we integrate to x from 0 to 1, because that's the only spot where there is area under the curve, with respect to x, we get x squared evaluated from 1 to 0, which is just 1 minus 0, which gives us 1.
03:34
So this is our valid probability density function.
03:38
So now we want to evaluate the probability is that x equals 0 .5.
03:44
And because this is a continuous probability density function, the probability of x equaling any one value is equal to zero, right? because to evaluate what the probability is we have to take the integral, from the lower bound to the upper bound.
04:14
So if we take the integral from 0 .5 to 0 .5 of f of x, this is just going to give us the same function evaluated at the same value, which equals 0.
04:30
However, we can do this for when x, the probability that x is less than two thirds, because now what we're going to be doing is we're going to be, what you can either do is you can integrate the density function, that we've already calculated, or since we're already given the cumulative distribution function, we know that the probability that x is less than or equal to two -thirds is just equal to big f of two -thirds, which is equal to, since x is between zero and one, x squared, which is equal to four ninths or point four for for repeating.
05:16
Now the probability that the absolute value of x is minus 0 .05 is less than or equal to 0 .1.
05:29
This is made unnecessarily complicated, so we can say that x minus 0 .05 is going to be between negative 0 .1 and positive 0 .1, which means that x is going to be between 1 point or sorry 0 .15 and negative 0 .5.
06:13
So for example, you can tell that this is an identical statement as the one above because if x equals point 0.
06:22
Sorry this should be negative 0 .05...