Difference between likelihood functions for pmf vs pdf Can someone explain the intuition behind how the likelihood function for a specific value of $\theta$ is different if $f_\theta$ is a pmf vs a pdf?
I thought that it was simply the probability that a certain outcome is observed, which is basically the pmf/pdf.

Simply, Likelihoods for discrete pmfs are probabilities, but for continuous pdfs, they are not. They go above 1. It is the area beneath them (over a tiny range) that is the probability.
Likelihood is a single point on a probability function. If that function is discrete, then the x axis is cut up into bands of a single integer, so each bin width is 1. So, the area is 1 times the value of the likelihood which is just the likelihood, so the likelihood is the probability. For a continuous pdf, to find that area you need integration.
