00:02
So we have to prove that the sum of observation of random sample of size in from an exponential distribution with parameter theta is a sufficient statistic for a theta so let it started to prove first of all the concept which we are going to use would be if x1 x2 up to that dash x of n be the random sample from the exponential population with parameter theta then probability mass function this can be given as f of y is equal to one over theta b to the power minus of x by theta were x greater than to 0 and theta greater than to 0.
01:08
This is what the definition, probability mass function and formula.
01:14
Now by differentiating the lycote function, which is given by l is equal to summation pi iota varying from 1 to n, f of x theta.
01:32
So basically this we can just write it down in pre -crant form like f of x1, dot, f of x2, dot dot dot dot, f of x t means if to up to that dash, f of x n.
01:51
See probability mass function we have just written here.
01:55
If we apply this for every and each of the terms, so for x1, it will be 1 over theta.
02:01
E to the power minus of x 1 by theta multiply 1 over theta 8 to the power minus x 2 by theta up to that does 1 over theta multiply by it to the power minus of x n over theta no see 1 by theta 1 by theta 1 by theta is just repeating so on up to that dash how much n so we can just say l is equal to 1 over theta is repeating n times as much as the series will goes on...