exercise:6eeb09869c: 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>S_N = X_1 + X_2 +\cdots+ X_N</math>, where the <math>X_i</math>'s are independent random variables with common distribution having generating function <math>f(z)</math>. Assume that <math>N</math> is an integer valued random variable in...") |
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Let <math>S_N = X_1 + X_2 +\cdots+ X_N</math>, where the <math>X_i</math>'s are independent random | |||
variables with common distribution having generating function <math>f(z)</math>. Assume that <math>N</math> is an integer | variables with common distribution having generating function <math>f(z)</math>. Assume that <math>N</math> is an integer | ||
valued random variable independent of all of the <math>X_j</math> and having generating function <math>g(z)</math>. | valued random variable independent of all of the <math>X_j</math> and having generating function <math>g(z)</math>. | ||
Latest revision as of 00:49, 15 June 2024
Let [math]S_N = X_1 + X_2 +\cdots+ X_N[/math], where the [math]X_i[/math]'s are independent random variables with common distribution having generating function [math]f(z)[/math]. Assume that [math]N[/math] is an integer valued random variable independent of all of the [math]X_j[/math] and having generating function [math]g(z)[/math]. Show that the generating function for [math]S_N[/math] is [math]h(z) = g(f(z))[/math]. Hint: Use the fact that
[[math]]
h(z) = E(z^{S_N}) = \sum_k E(z^{S_N} | N = k) P(N = k)\ .
[[/math]]