Estimation of an exponential parameter

Question

I´m trying to figure out the pdf $f_\min(X_i)$ of $\min(X_i)$, where the distribution of the sample $X_1,...,X_n$ is $\mathcal{E}xp(\lambda)$, where $\lambda$ is the unknown parameter.

I tried with the PDF of the exponential, and it gave me $\lambda*\exp(-\lambda x)$. But I don't know if its correct.

Can you help me out with this?

It was a typo. But I cannot find on wikipedia the minimum pdf for a exponential. — Ben C., Feb 25 '19 at 07:08
[Opening the Wikipedia page, I can find it...](https://en.wikipedia.org/wiki/Order_statistic#Order_statistics_sampled_from_an_exponential_distribution) — Xi'an, Feb 25 '19 at 07:16
See https://stats.stackexchange.com/q/306342/919 or https://stats.stackexchange.com/q/26029/919. — whuber, Feb 25 '19 at 13:51

score 2 · Answer 1 · answered Feb 25 '19 at 07:10

2

The answer can't be correct. We should expect the answer to be a function of $n$.

Let $Y=\min_i X_i$.

Then \begin{align} P(Y \le y) &= 1-P(Y > y)\\ &=1 - \prod_{i=1}^n P\left(X_i > y \right)\\ &=1 - \prod_{i=1}^n \left(1-F_{X_i}(y)\right)\\ &=1 - \left(1-F_{X_1}(y)\right)^n\\ \end{align}

Now, try to differentiate to get the answer.

answered Feb 25 '19 at 07:10

Siong Thye Goh

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And what would it be in terms of "x" and lambda?. Since I don´t want it to depend on "n". – Ben C. Feb 25 '19 at 07:14
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If something depends on $n$, you can't just ignore it. – Siong Thye Goh Feb 25 '19 at 07:21

Estimation of an exponential parameter

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