Grinstead-Snell-Probability · Problem 10.1.10

Generating Functions

Mathematics · generating-functions-discrete

Problem

Show that if

h(z) = \frac{1 - \sqrt{1 - 4pqz^2}}{2qz},

then

h(1) = \begin{cases} p/q, & \text{if } p \leq q, \\ 1, & \text{if } p \geq q, \end{cases}

and

h'(1) = \begin{cases} 1/(p-q), & \text{if } p > q, \\ \infty, & \text{if } p = q. \end{cases}