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Let $X_1, . . . , X_n$ be a random sample from the following distribution with parameter $\lambda$ and pdf:$f(x|\lambda ) = \lambda e^{−\lambda(x−2)}, x >2$.

I found the MLE to be $\displaystyle \lambda=\frac{n}{\sum x_i-2}$

The next question then asks if this estimator is unbiased.

How do I work on this?

Luis Alexandher
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mikaelaa
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  • Take $Y=X-2$ and compare with this question: https://math.stackexchange.com/q/2034206/321264. – StubbornAtom Nov 03 '21 at 17:00
  • Hi there! What would be the distribution for ∑X -2 – mikaelaa Nov 03 '21 at 17:15
  • @NicholasTan Why do you want to know this? It´s not needed for the exercise. But it is a gamma distribution. – callculus42 Nov 03 '21 at 17:25
  • Are you sure the MLE isn't $\hat\lambda=n/\sum_i(X_i-2)$?This denominator has a Gamma distribution, as is mentioned in the linked post. – StubbornAtom Nov 03 '21 at 17:59
  • Yes, but I am still confused. Would the range of feasibility be 2 to infinity when trying to find the expected value of the MLE. Also, what would be the parameters of ∑(−2) ~ Gamma ? – mikaelaa Nov 04 '21 at 00:33

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