EX 3.12 gender of newborn child

• Observation Setup
  • Start at a fixed time
  • Observe gender of each newborn child at a certain hospital
  • Continue until a boy (B) is born
• Definitions
  • Let $p=P(B)$
  • Assume that successive births are independent
  • Define the random variable (rv) $X$
    • $X$ = number of births observed

$$\begin{array}{rl}p(1)& =P(X=1)\\ & =P(B)=p\\ p(2)& =P(X=2)\\ & =P(GB)=P(G)\cdot P(B)=(1-p)p\\ p(3)& =P(X=3)\\ & =P(GGB)=P(G)\cdot P(G)\cdot P(B)\\ & =(1-p{)}^{2}p.\end{array}$$

• Continuing in this way, a general formula emerges: p\left( x\right) = \left\{ \begin{matrix} {\left( 1 - p\right) }^{x - 1}p & x = 1,2,3,\ldots \\ 0 & \text{ otherwise } \end{matrix}\right. \tag{3.2}
• Parameter $p$
  • Can assume any value between 0 and 1
• Geometric Distributions
  • Expression (3.2) describes the family of geometric distributions
• Specific Scenarios
  • Gender Scenario
    • Appropriate $p=0.51$
  • Rh-Positive Blood Scenario
    • Appropriate $p=0.85$