Explain the distribution function of a random variable.

Discrete distribution function

The discrete cumulative distribution function or distribution function of a real valued discrete random variable X takes the countable number of points x1, x2 , ...with corresponding probabilities p(x1), p(x2 ), ... and then the cumulative distribution function is defined by 

For instance, suppose we have a family of two children. The sample space 

S = {bb, bg, gb, gg}, where b =boy and g = girl

Let X be the random variable which counts the number of boys. Then, the values (X) corresponding to the sample space are 2, 1, 1, and 0.