exercise:543909e64c: 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> Consider a branching process such that the number of offspring of a parent is <math>j</math> with probability <math>1/2^{j + 1}</math> for <math>j = 0</math>, 1, 2, \ldots. <ul><li> Using the results of Example sh...") |
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Consider a branching process such that the number of offspring of a parent is <math>j</math> with probability <math>1/2^{j + 1}</math> for <math>j = 0</math>, 1, 2, .... | |||
<ul style="list-style-type:lower-alpha"><li> Using the results of [[guide:Baa5a33dd4#exam 10.2.4 |Example]] show that the probability that | |||
parent is <math>j</math> with probability <math>1/2^{j + 1}</math> for <math>j = 0</math>, 1, 2, | |||
<ul><li> Using the results of [[guide:Baa5a33dd4#exam 10.2.4 |Example]] show that the probability that | |||
there are <math>j</math> offspring in the <math>n</math>th generation is | there are <math>j</math> offspring in the <math>n</math>th generation is | ||
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p_j^{(n)} = \left \{ \begin{array}{ll} | p_j^{(n)} = \left \{ \begin{array}{ll} | ||
\frac{1}{n(n + 1)} (\frac {n}{n + 1})^j, & \mbox{if $ j \geq 1$}, \\ | \frac{1}{n(n + 1)} (\frac {n}{n + 1})^j, & \mbox{if $ j \geq 1$}, \\ | ||
\frac {n}{n + 1}, & \mbox{if | \frac {n}{n + 1}, & \mbox{if $ j = 0$}.\end{array}\right. | ||
</math> | </math> | ||
Latest revision as of 23:56, 14 June 2024
Consider a branching process such that the number of offspring of a parent is [math]j[/math] with probability [math]1/2^{j + 1}[/math] for [math]j = 0[/math], 1, 2, ....
- Using the results of Example show that the probability that
there are [math]j[/math] offspring in the [math]n[/math]th generation is
[[math]] p_j^{(n)} = \left \{ \begin{array}{ll} \frac{1}{n(n + 1)} (\frac {n}{n + 1})^j, & \mbox{if $ j \geq 1$}, \\ \frac {n}{n + 1}, & \mbox{if $ j = 0$}.\end{array}\right. [[/math]]
- Show that the probability that the process dies out exactly at the [math]n[/math]th generation is [math]1/n(n + 1)[/math].
- Show that the expected lifetime is infinite even though [math]d = 1[/math].