Ques 6- There are two continuous probability distributions: A is an exponential distribution with mean θ = 7 B is a distribution that is uniform on the interval from 0 to 8, and thereafter proportional to A. Show that the probability of a random variable that follows distribution B and lying between 0 and 8 is 8/15 Since my distribution function is uniform in the range 0-8, we should have fX(x) as (1\8) and not c. Soultion:- Please explain why have we taken c
The question says that in the interval [0,8] B has a uniform distribution and after that i.e. [8,infinity] B has a distribution which is proportional to A that has an exponential distribution. So Probability under all the entire range i.e [0, infinity] will be 1. Therefore the what u have considered is wrong.
I dont think so because they havent specified uniform distribution.They have just said that beta is uniform over a particular range. That means it is some constant which they have considered here as c. This is how they got 8c as they have integrated c over the range 0 to 8
If it is not so they could have easily taken 1/8 as PDF in the interval [0,8] as Beta follows uniform distribution instead they took 'c' bcoz the total probability under interval [0,8] is unkown and hence it is assumed c. Moreover, you can see that later they have taken total probability under [0,8] plus under [0,infinity] as 1.
