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