# Seeing if estimators are unbiased

I have the pdf

$$f(y ; \theta) = \frac{1}{\theta} \exp( \frac{-y}{\theta}), \ y > 0$$

and I'm supposed to determine if the following two estimators are unbiased or not: $\hat \theta = nY_{min}$ and $\hat \theta = \frac{1}{n}\sum_{i=1}^n Y_i$. I'm running into some problems because when I try to find the expected value of $Y_{min}$ and $Y_i$, the integral is undefined.

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Please see the following Wikipedia article and it shows that both the integral exist. Also, if you look lower in the article, you will also see that both the estimators are unbiased.

$Y_{min}$ is also exponentially distributed with mean parameter $\frac {\theta} {n}$

Therefore, $E[nY_{min}]=n\times\frac {\theta} {n} = \theta$. Similarly, $E[\frac{1}{n}\sum_{i=1}^n Y_i] = \frac{n \theta}{n}=\theta$

I am sure both the integrals exists

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Ya, realized what I was doing. Missed that whole exponential thing. Thanks. –  tshauck Feb 9 '11 at 4:28