exercise:7a48130c99: Difference between revisions
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Let <math>X</math> be the first time that a ''failure'' occurs in an infinite sequence of Bernoulli trials with probability <math>p</math> for success. Let <math>p_k= P(X = k)</math> for <math>k = 1</math>, 2, | Let <math>X</math> be the first time that a ''failure'' occurs in an infinite sequence of Bernoulli trials with probability <math>p</math> for success. Let <math>p_k= P(X = k)</math> for <math>k = 1</math>, 2, .... Show that <math>p_k = p^{k - 1}q</math> where <math>q = 1 - p</math>. | ||
Show that <math>\sum_k p_k = 1</math>. Show that <math>E(X) = 1/q</math>. What is the expected number of tosses of a coin required to obtain the first tail? | Show that <math>\sum_k p_k = 1</math>. Show that <math>E(X) = 1/q</math>. What is the expected number of tosses of a coin required to obtain the first tail? | ||
Latest revision as of 19:16, 24 June 2024
Let [math]X[/math] be the first time that a failure occurs in an infinite sequence of Bernoulli trials with probability [math]p[/math] for success. Let [math]p_k= P(X = k)[/math] for [math]k = 1[/math], 2, .... Show that [math]p_k = p^{k - 1}q[/math] where [math]q = 1 - p[/math]. Show that [math]\sum_k p_k = 1[/math]. Show that [math]E(X) = 1/q[/math]. What is the expected number of tosses of a coin required to obtain the first tail?