BBy Bot
Jun 09'24
Exercise
[math]
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Show that [math]w_x[/math] of Exercise satisfies the following conditions:
- [math]w_x = pw_{x + 1} + qw_{x - 1}[/math] for [math]x = 1[/math], 2, ..., [math]T - 1[/math].
- [math]w_0 = 0[/math].
- [math]w_T = 1[/math].
Show that these conditions determine [math]w_x[/math]. Show that, if [math]p = q =1/2[/math], then
[[math]]
w_x = \frac xT
[[/math]]
satisfies (a), (b), and (c) and hence is the solution. If [math]p \ne q[/math], show that
[[math]]
w_x = \frac{(q/p)^x - 1}{(q/p)^T - 1}
[[/math]]
satisfies these conditions and hence gives the probability of the gambler winning.