exercise:Ff5a3bc8f7: 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> People arrive at a queue according to the following scheme: During each minute of time either 0 or 1 person arrives. The probability that 1 person arrives is <math>p</math> and that no person arrives is <math>q = 1 - p</math>. Let <math>C_r</ma...") |
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People arrive at a queue according to the following scheme: During each minute of time either 0 or 1 person arrives. The probability that 1 person | |||
minute of time either 0 or 1 person arrives. The probability that 1 person | |||
arrives is <math>p</math> and that no person arrives is <math>q = 1 - p</math>. Let <math>C_r</math> be the | arrives is <math>p</math> and that no person arrives is <math>q = 1 - p</math>. Let <math>C_r</math> be the | ||
number of customers arriving in the first <math>r</math> minutes. Consider a Bernoulli | number of customers arriving in the first <math>r</math> minutes. Consider a Bernoulli | ||
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if no person arrives in a unit time. Let <math>T_r</math> be the number of failures | if no person arrives in a unit time. Let <math>T_r</math> be the number of failures | ||
before the <math>r</math>th success. | before the <math>r</math>th success. | ||
<ul><li> What is the distribution for <math>T_r</math>? | <ul style="list-style-type:lower-alpha"><li> What is the distribution for <math>T_r</math>? | ||
</li> | </li> | ||
<li> What is the distribution for <math>C_r</math>? | <li> What is the distribution for <math>C_r</math>? | ||
Latest revision as of 23:58, 24 June 2024
People arrive at a queue according to the following scheme: During each minute of time either 0 or 1 person arrives. The probability that 1 person arrives is [math]p[/math] and that no person arrives is [math]q = 1 - p[/math]. Let [math]C_r[/math] be the number of customers arriving in the first [math]r[/math] minutes. Consider a Bernoulli trials process with a success if a person arrives in a unit time and failure if no person arrives in a unit time. Let [math]T_r[/math] be the number of failures before the [math]r[/math]th success.
- What is the distribution for [math]T_r[/math]?
- What is the distribution for [math]C_r[/math]?
- Find the mean and variance for the number of customers arriving in the first [math]r[/math] minutes.