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I am learning about the likelihood function given iid random variables $X_i$ and realizations $x_i$: $\mathcal{L}(\theta | x) = \prod_{i=1}^n \mathbb{P}(X_i = x_i)$. One thing I am confused about is the $\mathbb{P}(X_i = x_i)$ expression. For probability density functions, isn't this not well defined? My understanding is that a PDF (continuous) is used for measuring area under the curve between two points and can thus be negative or zero at fixed points since the area is 0 at a fixed point (and has a probability of 0 of being a specific number). I must be misunderstanding something.

timeinbaku
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    Related: https://stats.stackexchange.com/questions/243283/in-mle-for-continuous-rv-why-is-it-ok-to-evaluate-a-pdf-at-a-point/243692#243692 – Zhanxiong Jul 08 '23 at 18:32
  • Thanks @Zhanxiong, this answers my question! – timeinbaku Jul 08 '23 at 18:39
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    A density function is well-defined almost everywhere, which includes the observed realisation of the random sample. – Xi'an Jul 08 '23 at 20:08

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