What is the CDF of a discrete random variable? Is there an explicit formula of the CDF of a discrete random variable? I know that a CDF of a continuous (real-valued) random variable is:
$$F_X(x)=\Pr[X\leq x]$$
Is there an equivalent formula for discrete random variable?
Best Answer
$F_X(x)=\Pr[X\leq x]$ is the definition of a cumulative distribution function, whether the random variable has a discrete or a continuous distribution.
For a discrete random variable you can write $$F_X(x)=\Pr[X\leq x] = \sum_{y \leq x}\Pr[X= y]$$ while for a continuous random variable with a probability density function $f_X$ it could be $$F_X(x)=\Pr[X\leq x] = \int_{-\infty}^x f_X(y)\; dy$$