# Struggling with proving an inconsistent estimator: Stat Theory

So I have this homework problem that goes: Let X~exp(λ), with λ unknown, and let $X_1...X_n$ be a sample on X. Show that T=$\sum X_i$ is not a consistent estimator for λ.

I know that the consistent estimator formula is P(|$X_n$-c|>$\epsilon$) Then my formula becomes:

P(|$\sum X_i-λ|>\epsilon$)

This is where I get stuck because I know in order for an estimator to be convergent, the whole equation needs to go to 0. I know that since we are not dealing with the mean of the exponential 1/λ and the proportional mean that this is never going to converge to 0. I am just not sure how I am going to say this. I know I can possibly go about this the following way:

Let $\epsilon$=$λ^2$/λ

P(|$\sum X_i-|>λ^2$/λ)

= P($\sum X_i-λ>λ^2+λ^2$/λ) + P($\sum X_i-λ<λ^2-λ^2$/λ)

= P($\sum X_i-λ>λ^2+λ^2$/λ) + 0 = $e^(-λ(λ+λ^2/λ))$ = $e^(-λ^2-λ^2)$ = $e^(2λ)$ And this converges to 0, which does not disprove the consistency.

I am not sure what at this point is going to make this not consistent.

Could someone let me know any thoughts on how to make this inconsistent?

Thanks!!

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