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If I wanted to maximum likelihood estimator for $f(x)=\frac{1}{\sqrt{\sigma^22\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$ where $-\infty\leq x \leq \infty$, my original plan was to do $\Pi^{n=\infty}f(x)$ but that really doesn't make any sense. The confusion arises since when you usually find the MLE, we' do $\Pi^{n}_{i=1}f(x)$ but I'm given the bound $-\infty\leq x \leq \infty$ so I can't just simply do the old trick of doing $\ln(L(\theta)) = ...$ and taking the derivative since there's going to be INFINITE terms

Other similar questions asked on this stackExchange has been bounded by some $n$ but what if it isn't?

John Rawls
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  • Why would you need an integral? Or, more directly, how is the MLE defined? – Sycorax Mar 06 '21 at 00:57
  • @Sycorax hm I guessed an integral because it's continuous where there's an infinite amount of X's. its easy to get the MLE for this if there were N random variables but there isn't - unless if I'm misunderstanding something – John Rawls Mar 06 '21 at 01:10
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    The maximum likelihood estimator is the estimator that maximizes the joint density (or joint probability) of the sample. If you have finite data, then you don't have "an infinite amount of X's." This is explained in more detail in the duplicate threads. Once you've read them, you can edit your question to clarify what you'd like to know and we can reopen the question. – Sycorax Mar 06 '21 at 01:15
  • @Sycorax alright just edited – John Rawls Mar 06 '21 at 01:28
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    I don't see an integral anywhere in this problem. I also don't see where these 'infinite terms' are coming from. Are you trying to say that every single possible $x \in \mathbb{R}$ is a data point for you? – Arya McCarthy Mar 06 '21 at 02:42
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    I've read the edited question, but it's still not clear how you've arrived at "infinite" anything. The normal distribution assigns a positive density to every real number -- is this the thing that's supposedly infinite in your expression? Because the support of a normal distribution doesn't have much bearing on how to write down the MLE for a normal distribution. – Sycorax Mar 06 '21 at 03:05

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