exercise:99a7c13fba: 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> Let <math>p</math> be the probability distribution <math display="block"> p = \pmatrix{ 0 & 1 & 2 \cr 0 & 1/3 & 2/3 \cr}\ , </math> and let <math>p_n = p * p * \cdots * p</math> be the <math>n</math>-fold convolution of <math>p</math> with itself...") |
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Let <math>p</math> be the probability distribution | |||
<math display="block"> | <math display="block"> | ||
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and let <math>p_n = p * p * \cdots * p</math> be the <math>n</math>-fold convolution of <math>p</math> with | and let <math>p_n = p * p * \cdots * p</math> be the <math>n</math>-fold convolution of <math>p</math> with | ||
itself. | itself. | ||
<ul><li> Find <math>p_2</math> by direct calculation (see [[guide:4c79910a98#defn 7.1 |Definition]]). | <ul style="list-style-type:lower-alpha"><li> Find <math>p_2</math> by direct calculation (see [[guide:4c79910a98#defn 7.1 |Definition]]). | ||
</li> | </li> | ||
<li> Find the ordinary generating functions <math>h(z)</math> and <math>h_2(z)</math> for | <li> Find the ordinary generating functions <math>h(z)</math> and <math>h_2(z)</math> for | ||
Latest revision as of 00:44, 15 June 2024
Let [math]p[/math] be the probability distribution
[[math]]
p = \pmatrix{
0 & 1 & 2 \cr
0 & 1/3 & 2/3 \cr}\ ,
[[/math]]
and let [math]p_n = p * p * \cdots * p[/math] be the [math]n[/math]-fold convolution of [math]p[/math] with itself.
- Find [math]p_2[/math] by direct calculation (see Definition).
- Find the ordinary generating functions [math]h(z)[/math] and [math]h_2(z)[/math] for [math]p[/math] and [math]p_2[/math], and verify that [math]h_2(z) = (h(z))^2[/math].
- Find [math]h_n(z)[/math] from [math]h(z)[/math].
- Find the first two moments, and hence the mean and variance, of [math]p_n[/math] from [math]h_n(z)[/math]. Verify that the mean of [math]p_n[/math] is [math]n[/math] times the mean of [math]p[/math].
- Find those integers [math]j[/math] for which [math]p_n(j) \gt 0[/math] from [math]h_n(z)[/math].