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)