00:01
In this question, the f -of -x is given here, and the f -fax is the probability -dance -the -function.
00:06
So in the first part, we have to find the value of c.
00:09
So the probability of dense -the -function, the integral value should be equal to 0.
00:13
So from a to b, f -of -x, the x, should be equal to 1 or probability -density function.
00:23
Let's use this rule to find the value of c here.
00:27
So between negative 1 and 1, there is c times 1 minus x squared, the x.
00:36
So that should be c times x minus x cubed over 3.
00:41
Let's plug in the values negative 1 and 1.
00:45
So we will get c times this is 1 minus 1 over 3 minus.
00:50
This is negative 1 plus 1 over 3.
00:53
If i simplify them, we will get c times this is 4 over 3, which is 1.
00:59
From here c is equal to 3 over 4 so we got the value of c here and in the next part so the question is being asked find the probability of 0 x and 1 half so when we'll get this piecewise function that is between the interval negative 1 and 1 so let me just take the integral of this value from this values 0 to 1 1 1 1 1 1 1⁄2 this is 3 over 4 1 minus x squared because we got the c value as 3 over 4.
01:32
So let's take the integral of this value, which is x minus x cubed over 3, and the integral values are 0 and 1 half.
01:42
If i just plug in the values here, so we will get 3 over 4 times.
01:48
This is 1 1⁄2, 1⁄2, 1⁄2, 2 and 4.
01:54
So if you just do this calculation, we will get 11 over 32.
01:58
So this is the probability for this.
02:01
Given function for the given interval.
02:04
And the next one, in the third part, we have to find the variance...