EX 3.19 first born boy

The general form for the pmf of $X=$ the number of children born up to and including the first boy is $p\left(x\right)=\{\begin{array}{cc}p{\left(1-p\right)}^{x-1}& x=1,2,3,\dots \\ 0& \text{ otherwise }\end{array}$

From the definition,

$$\begin{array}{rl}E(X)& =\sum _{D}x\cdot p(x)\\ & =\sum _{x=1}^{\infty }xp(1-p{)}^{x-1}\\ & =p\sum _{x=1}^{\infty }\left[-\frac{d}{dp}(1-p{)}^x\right]\end{array}\text{\hspace{1em}(3.9)}$$

Interchanging the order of taking the derivative and the summation, the sum is that of a geometric series.

After the sum is computed, the derivative is taken, resulting in $E\left(X\right)=1/p$ .

• If $p$ is near 1, we expect to see a boy very soon,
• whereas if $p$ is near 0, we expect many births before the first boy.

For $p=.5$, $E\left(X\right)=2$.