3. Suppose that X is a continuous random variable whose...

3. Suppose that X is a continuous random variable whose probability density function is given by: f(x) = { C(1 - x^2) for -1 < x < 1 { 0 otherwise (1) What is the value of C? (2) Find P(0 < X < 1/2). (3) Find the variance Var(X). (15%)

Transcript

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...

Question

3. Suppose that X is a continuous random variable whose probability density function is given by: f(x) = { C(1 - x^2) for -1 < x < 1 { 0 otherwise (1) What is the value of C? (2) Find P(0 < X < 1/2). (3) Find the variance Var(X). (15%)

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