Problem 2 Let X be a random variable, defined by its...

Problem 2 Let X be a random variable, defined by its probability density function f_X(x) = { x if 0 < x < 1 2 - x if 1 < x < 2 0 else. (a) Let events A and B be defined as A = {X < 1} and B = {1/2 < X < 3/2}. Show that A and B are independent. (b) Define a new random variable Y = X^3. Find the probability distribution function F_Y(y).

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0:00 Hello students.

00:01 Today we will discuss about this question.

00:04 In this question we are given that x be a random variable defined its probability density function that is fx of x that is equals to x if 0 x is less than 1, 2 minus x if 1 is less than x is less than 2 and 0 else.

00:28 And we are given event a and b that can be defined as a that is equals to x is less than 1 and b that is equals to 1 divided by 2 is less than x is less than 3 divide by 2.

00:46 And we need to show that a and b are independent event.

00:56 So here for that event a that can be given as so here first of all we need to to find a and b.

01:07 So that is equals to we will get 1 divided by 2 is less than x is less than 1 because the interval must be common in both events a and b.

01:19 So therefore p of a that is equals to integration 0 to 1 fx of x d x that is equals to integration 0 to 1 x d x that is equals to x squared divided by 2 and the limit will be 0 to 1 so that is equals to 1 divided by 2 minus 0 so that is equals to 1 divide by 2 therefore p of a that is equals to half now we will find the p of b so that is equal to probability of 1 divide by 2 is less than x is less than 3 divided by 2 that is equals to integration 1 divide by 2 to 1 f x d x plus 1 2 2 2 2 of x d x plus 1 2 2 2, 3 divided by 2, fx of x d x.

02:05 So that is equals to integration 1 divided by 2 to 1 x d x plus integration 1 to 3 divide by 2 minus x d x.

02:17 So that is equals to x squared divide by 2 limit will be minus 1 by 2 to 1 plus 2x minus x square divide by 2, limit will be 1, 2 3 divided by 2...

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