7. A discrete random variable X has a pdf of the form f(x) =...

7. A discrete random variable X has a pdf of the form f(x) = c(8 - x) for x = 0, 1, 2, 3, 4, 5, and zero otherwise. (a) Find the constant c. (b) Find the CDF, F(x). (c) Find P[X > 2]. (d) Find E(X).

Transcript

00:01 Hello and welcome to this video solution of numerate.

00:04 So here a discrete random variable x has a probability density function in the form of x that is equal to c times of 8 minus x for x equal to 0 1 to 3 4 and 5 and 0 otherwise okay.

00:19 Now we define the constant c pretty easy here.

00:23 So a you know the summation of this function of x from 0 to 5 right as per this shall be equal to 1 itself.

00:33 So you know that is summation of 0 to 5 the function is c 8 minus x that is equal to 1 and now you have to just plug in x equal to sorry x equal to 0 to 5 here so that gives you c times of 8 minus 0 plus c times of 8 minus 1 plus c times of 8 minus 2 plus c times of 8 minus 2 plus c times of 8 minus 3 c times of 8 minus 4 and c times of 8 minus 5 that is equal to 1 right and you have got that is c to be equal to 1 by 33 right next be you have to find out the cumulative distribution function right b so that is given by f x that is equal to you can consider it summation of x from k equal to 0 to f k right and you know this f x here clearly is equal to 8 minus x by 33 you have to consider now here if 0 is equal to yeah f of 0 itself that's from here 8 minus 0 by 33 that is equal to 8 by 33 you have got now f1 will be equal to f0 plus f 1 right that is 8 by 33 plus here you have got 7 by 33 right so like that so this is equal to 5 by 11 now f2 you can say that it's f you can say this is actually if capital of 0 not small f f 0 this is capital f 1 plus f of 2 right so that is 5 by 11 right so that is 5 by 11 plus here you have got 6 by 33 that is 7 by 11 now f of 3 is f2 plus f3 itself and that is equal to 7 by 11 plus f3 you know f3 is 5 by 33 that is equal to 26 by 33 f4 we have got f3 plus f4 f3 is 26 by 33 plus f4 f4 is 4 by 33 which is equal to 10 by 11 now finally f5 is equal to f4 plus f5 right so f4 is 10 by 11 plus this is 3 by 33 which is equal to 1 so this is what we have as a cumulative distributive function.

03:53 Or maybe if you want to write down fx, then you have to take it like, yeah, 0 for x less than 0...

Question

7. A discrete random variable X has a pdf of the form f(x) = c(8 - x) for x = 0, 1, 2, 3, 4, 5, and zero otherwise. (a) Find the constant c. (b) Find the CDF, F(x). (c) Find P[X > 2]. (d) Find E(X).

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