probability density function — Jemoka Knowledge Base

PDFs is a function that maps continuous random variables to the corresponding probability.

\begin{equation} P(a < X < b) = \int_{x=a}^{b} f(X=x)\dd{x} \end{equation}

note: f is no longer in units of probability!!! it is in units of probability scaled by units of X. That is, they are DERIVATIVES of probabilities. That is, the units of f should be \frac{prob}{unit\ X}. So, it can be greater than 1. We have two important properties: if you integrate over any bounds over a probability density function, you get a probability if you integrate over infinity, the result should be 1 getting exact values from PDF There is a calculus definition for P(X=x), if absolutely needed:

\begin{equation} P(X=x) = \epsilon f(x) \end{equation}

mixing discrete and continuous random variables Let’s say X is continuous, and N is discrete. We desire:

\begin{equation} P(N=n|X=x) = \frac{P(X=x|N=n)P(N=n)}{P(X=x)} \end{equation}

now, to get a specific value for P(X=x), we can just multiply its PMF by a small epsilon:

\begin{align} P(N=n|X=x) &= \lim_{\epsilon \to 0} \frac{\epsilon f(X=x|N=n)P(N=n)}{\epsilon f(X=x)} \\ &= \frac{f(X=x|N=n)P(N=n)}{f(X=x)} \end{align}

this same trick works pretty much everywhere—whenever we need to get the probability of a continuous random variable with