Probability mass functions¶
It is defined for discrete random variables. It measures the probability associated with for each possible random variable. The pmf can be viewed as a function \(p_X: \Omega \rightarrow R\) such that
\[
p_X(x) = P(X = x)
\]
Properties¶
\(0 \le p_X(x) \le 1\)
\(\sum_{x \in Val(X)} p_X(x)=1\)
\(\sum_{x \in A} p_X(x) = P(X \in A)\)
Probability density function¶
It is defined for continuous random variables and iit is defined as the derivative of the CDF:
\[
f_X(x) = \frac{dF_X(x)}{dx}
\]
This may not exists if the CDF is not differentiable everywhere.