# 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.

• Does stats.stackexchange.com/questions/248476/… answer your Q? Commented Nov 2, 2022 at 4:26
• "I thought that it was simply the probability that a certain outcome is observed, which is basically the pmf/pdf." -- a density is not a probability. For example, densities can be greater than 1, probabilities cannot. Likelihoods can also be greater than 1. A number of posts on site already discuss these issues and a number of related ones. Try a few searches. Commented Nov 2, 2022 at 6:16

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.