BBy Bot
Jun 09'24
Exercise
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Assume in the gambler's ruin problem that [math]p = q = 1/2[/math].
- Using Equation, together with the facts that [math]q_0 = 1[/math] and [math]q_M = 0[/math],
show that for [math]0 \le z \le M[/math],
[[math]] q_z = {{M - z}\over M}\ . [[/math]]
- In Equation, let [math]p \rightarrow 1/2[/math] (and since [math]q = 1 - p[/math], [math]q
\rightarrow 1/2[/math] as well). Show that in the limit,
[[math]] q_z = {{M - z}\over M}\ . [[/math]]Hint: Replace [math]q[/math] by [math]1-p[/math], and use L'Hopital's rule.