1. A certain random variable X has the continuous...

1. A certain random variable X has the continuous distribution with p.d.f. (f(x) = Cx^2 (1 - x^2)^3), if (I(-1 < x < 1)), where (I(.)) is an indicator function. Then, (a) Evaluate C the normalizing constant. [3] (b) Obtain the distribution function of X, i.e., (F(x) = P(X le x)). [2] (c) What is (P(frac{1}{3} < X < frac{5}{4}))? [2]

Transcript

00:01 Hello students, we have given the pdf of the continuous random variable x which is f of x is equal to c into x square into 1 minus x square bracket cube.

00:13 Now this is equal to c into x square and now 1 minus x square bracket cube we can write it as 1 minus x raised to 6 minus 3 x square into 1 minus x square and now this is equal to c x square into bracket 1 minus x raised to 6 minus 3 x square plus 3 x raised to 4 and the range of x is from minus 1 to plus 1.

00:46 Now this is equal to c into bracket x square minus x raised to 8 minus 3 x raised to 4 plus 3 x raised to 6 and minus 1 is less than x is less than 1.

01:00 Now we have to find the value of c that is constant.

01:03 Now we know if f of x is a proper probability density function then we can write integration from minus 1 to plus 1 f of x into dx is equal to 1.

01:17 So this is equal to c into integration from minus 1 to plus 1 x square minus x raised to 8 minus 3 into x raised to 4 plus 3 into x raised to 6 into dx is equal to 1 that is equal to c into x cube divided by 3.

01:38 This is the integration of x square then integration of x raised to 8 is x raised to 9 divided by 9 minus 3 into x raised to 4 is x raised to 5 divided by 5 plus 3 into x raised to 7 divided by 7 and limit from minus 1 to plus 1 this is equal to 1.

02:01 So after putting the limits that is upper limit minus lower limit we will get c into 32 divided by 315 is equal to 1 and therefore the value of c that is constant is equal to 315 divided by 32.

02:21 Then next we have to obtain the distribution function that is capital f of x.

02:28 So we know the formula for the cdf that is capital f of x is equal to probability of x is less than or equal to x and that is it equal to integration from 1 to x f of t into dt.

02:48 So now this is equal to integration from 1 to x f of t into dt is 315 divided by 32 which is the value of constant c then t square into 1 minus t square bracket cube dt...

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