Question

Suppose that (X1, X2, . . . , Xn) is a random sample from a very large population (population size N n). What is the probability distributions of the first observation X1?

Answer 1

this is the case of simple random sampling , to find the probability distribution of X1 , let us first findout the probabilty of getting this sample selected from the population of size N

there can be total of of ^Nc_n number of samples from the population of size N , the probabilty of getting this sample selected is 1/ ^Nc_n , now there are n total observation in this sample , so the probability of getting X1 selected is

1/ ^Nc_n x 1/n (by the theorem of total probabilty)

probability of getting X1 selected is = 1/ ^Nc_n x 1/n