exercise:026effa772: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> (The next three problems come from Feller.<ref group="Notes" >W. Feller, op.\ cit., pg. 367.</ref>) As in the text, assume that <math>M</math> is a fixed positive integer. <ul><li> Show that if a gambler starts with an stake of 0 (and is allowed...") |
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\newcommand{\mathds}{\mathbb}</math></div> (The next three problems come from Feller.<ref group="Notes" >W. Feller, op. | \newcommand{\mathds}{\mathbb}</math></div> (The next three problems come from Feller.<ref group="Notes" >W. Feller, op. cit., | ||
pg. 367.</ref>) | pg. 367.</ref>) | ||
As in the text, assume that <math>M</math> is a fixed positive integer. | As in the text, assume that <math>M</math> is a fixed positive integer. | ||
<ul><li> Show that if a gambler starts with an stake of 0 (and is allowed to have a negative | <ul style="list-style-type:lower-alpha"><li> Show that if a gambler starts with an stake of 0 (and is allowed to have a negative | ||
amount of money), then the probability that her stake reaches the value of <math>M</math> before it returns | amount of money), then the probability that her stake reaches the value of <math>M</math> before it returns | ||
to 0 equals <math>p(1 - q_1)</math>. | to 0 equals <math>p(1 - q_1)</math>. | ||
Latest revision as of 01:56, 15 June 2024
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(The next three problems come from Feller.[Notes 1])
As in the text, assume that [math]M[/math] is a fixed positive integer.
- Show that if a gambler starts with an stake of 0 (and is allowed to have a negative amount of money), then the probability that her stake reaches the value of [math]M[/math] before it returns to 0 equals [math]p(1 - q_1)[/math].
- Show that if the gambler starts with a stake of [math]M[/math] then the probability that her stake reaches 0 before it returns to [math]M[/math] equals [math]qq_{M-1}[/math].
Notes