I am looking through notes for confidence interval for the exponential
mean.
I have been given that:
Suppose Y_1,...,Y_n ~ Exp(t^(-1)) independently. Then for each of the
Y_i:
f(y_i|t) = t^(-1) exp(-y_i/t).
We are then finding the maximum likelihood estimator ^t^, and with
various calculations (including logarithmic differentiation) we arrive
at the sample mean:
_
y (y-bar) = ^t^.
Now it is this part I don't understand:
The sampling distribution of ^t^ (t-hat) is obtained as follows:
Since Y_i ~ Exp(t^(-1))
Y_i/t ~ Exp(1) ==> Sum Y_i/t ~ Gamma(n,1), or:
n ^t^
----- ~ Gamma(n,1).
t
I just can't seem to follow the logic here. Can someone explain to me
what is going on?
Thanks,
Neeraj.
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