Discrete random variable: probability mass function versus cumulative distribution function

Summary

• \(X\) is a discrete random variable with probability mass function \(f\) , \(f\left( x \right)=P\left( X=x \right)\) , and cumulative distribution function \(F\) , \(F\left( x \right)=P\left( X≤x \right)\) .
• We define \( F\left( a- \right)\) as the left-hand limit

\[F\left( a- \right)=\lim_{x→a-} F(x)=P(X<a)\]

• The relationship between the pmf and the cdf is as follows:

\[f\left( x \right)=F\left( x \right)-F(x-)\]

\[F\left( x \right)=\sum_{i:x_i≤x}{ f\left( x_i \right) }\]