Cumulative distribution function¶
It is a function \(F_X: R \rightarrow [0,1]\) which sprecifies a probability measure:
\[
F_X(x) = P(X \le x)
\]
This is the probability that a random variable will be less then \(x\). We can use it to calculate the probability that a random variable will be within some bounds \(a,b\) where \(a < b\).
Properties¶
\(0 \le F_X(x) \le 1\)
\(\lim_{x \rightarrow - \infty} F_X(x) = 0\)
\(\lim_{x \rightarrow \infty} F_X(x) = 1\)
\(x \le y \Leftrightarrow F_X(x) \le F_X(y)\)