exercise:D793d2a774: Difference between revisions
(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> Suppose that <math>X</math> is a random variable which represents the number of calls coming in to a police station in a one-minute interval. In the text, we showed that <math>X</math> could be modelled using a Poisson distribution with parameter...") |
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Suppose that <math>X</math> is a random variable which represents the number of calls coming in to a police station in a one-minute interval. In the text, we showed that <math>X</math> could be modelled using a Poisson distribution with parameter | |||
number of calls coming in to a police station in a one-minute interval. In the text, | |||
we showed that <math>X</math> could be modelled using a Poisson distribution with parameter | |||
<math>\lambda</math>, where this parameter represents the average number of incoming calls per | <math>\lambda</math>, where this parameter represents the average number of incoming calls per | ||
minute. Now suppose that <math>Y</math> is a random variable which represents the number of | minute. Now suppose that <math>Y</math> is a random variable which represents the number of | ||
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P(Y = k) = e^{-\lambda t}{{(\lambda t)^k}\over{k!}}\ , | P(Y = k) = e^{-\lambda t}{{(\lambda t)^k}\over{k!}}\ , | ||
</math> | </math> | ||
i.e., <math>Y</math> is Poisson with | |||
parameter <math>\lambda t</math>. '' Hint'': Suppose a Martian were to observe the police | parameter <math>\lambda t</math>. '' Hint'': Suppose a Martian were to observe the police | ||
station. Let us also assume that the basic time interval used on Mars is exactly <math>t</math> | station. Let us also assume that the basic time interval used on Mars is exactly <math>t</math> | ||
Latest revision as of 00:04, 14 June 2024
Suppose that [math]X[/math] is a random variable which represents the number of calls coming in to a police station in a one-minute interval. In the text, we showed that [math]X[/math] could be modelled using a Poisson distribution with parameter [math]\lambda[/math], where this parameter represents the average number of incoming calls per minute. Now suppose that [math]Y[/math] is a random variable which represents the number of incoming calls in an interval of length [math]t[/math]. Show that the distribution of [math]Y[/math] is given by
i.e., [math]Y[/math] is Poisson with parameter [math]\lambda t[/math]. Hint: Suppose a Martian were to observe the police station. Let us also assume that the basic time interval used on Mars is exactly [math]t[/math] Earth minutes. Finally, we will assume that the Martian understands the derivation of the Poisson distribution in the text. What would she write down for the distribution of [math]Y[/math]?