00:01
Okay, so in this function, we're given this function, in this problem we're given this function f defined with a parameter k in here, when x is between 0 and 3, sorry, let me say here, and 0 otherwise.
00:13
And we want to answer three questions.
00:15
The first one is we want to find k such that this function represents a probability density function.
00:21
B, we want to find a probability that the random variable represented with the distribution like this is written equal to 2 and c the mean of such a random variable, right? so you just have to remember what the properties of probability density functions are.
00:34
They must be non -negative.
00:35
This one is always non -negative if k is greater than equal to zero.
00:38
And it must integrate to one.
00:41
Right.
00:41
So essentially we have to find k such that the integral over r of f -fx, the x, d x, is equal to one.
00:48
So let's make the calculation and then we equal the amount to one.
00:52
This will be k.
00:54
Of course, because it's zero outside of this interval, it's enough to integrate between 0 and 3, the function x squared 3 minus x, the x, right? so let's just write what this is.
01:05
So this would be 3x squared minus x cubed, the x from where we get, and we can integrate immediately, right? so the integral of 3x squared is x cubed.
01:18
The integral of minus x cubed will be minus x to the 4 over 4 between 0 and 3.
01:24
At 0, this will be 0.
01:25
So we simply get this value at 3 .3 cubed 27.
01:30
3 to the 4 is 81 over 4.
01:35
Let's just put everything to the same denominator.
01:37
4 times 27 is 80 plus 28.
01:41
So 108 minus 81 over 4.
01:46
And this gives us 27k over 4.
01:50
So the conclusion is that the interval is equal to 1.
01:53
If and only if k is, we take the inverse.
01:57
So 4 over 27.
02:00
So this is a case such that that function is a probability density function.
02:05
Second question, we want to find the probability that a random variable with this distribution is written and equal to 2.
02:12
And let's just remember that the probability that something is great and equal to 2 when we have density function would correspond to the area under the integral.
02:20
So this would be the integral between 2 and plus infinity of f of x the x, because f is a density function, right? so probabilities correspond to areas under the density function.
02:31
Okay.
02:33
And again, this is simply between two and three.
02:35
K is four over 27.
02:38
And then we have x squared, 1 minus x d x.
02:42
So let's make this calculation...