Cumulative probabilities

The cumulative probability for any value \(x\) is

\[P(X \le x) = \sum_{u \le x} p(u)\]

The cumulative distribution function generalises this:

Definition

The cumulative distribution function (CDF) for \(X\) is the function

\[F(x) = P(X \le x) = \sum_{u \le x} p(u)\]

The CDF is a step function, satisfying

\[ \begin{align} F(-\infty) &= 0\\ F(+\infty) &= 1 \end{align} \]

and increasing by \(p(x)\) at each \(x\).

Question

A couple want at least two children and no more than four, but will stop when they get a boy. Assuming that the probability of each child being a girl is \(\frac {1} {2} \), independently of the genders of previous children, the probability function for the number of girls in the resulting family is

Number of girls, x 0 1 2 3 4
p(x) 0.25 0.5 0.125 0.0625 0.0625

Draw the cumulative distribution function for X.

(Solved in full version)