exercise:F925535ee7: 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>X_1</math>, <math>X_2</math>, \ldots, <math>X_n</math> be an independent trials process with density <math display="block"> f(x) = \frac12 e^{-|x|}, \qquad -\infty < x < +\infty\ . </math> <ul><li> Find the mean and variance of <mat...") |
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Let <math>X_1</math>, <math>X_2</math>, ..., <math>X_n</math> be an independent trials process with | |||
density | density | ||
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f(x) = \frac12 e^{-|x|}, \qquad -\infty < x < +\infty\ . | f(x) = \frac12 e^{-|x|}, \qquad -\infty < x < +\infty\ . | ||
</math> | </math> | ||
<ul><li> Find the mean and variance of <math>f(x)</math>. | <ul style="list-style-type:lower-alpha"><li> Find the mean and variance of <math>f(x)</math>. | ||
</li> | </li> | ||
<li> Find the moment generating function for <math>X_1</math>, <math>S_n</math>, <math>A_n</math>, | <li> Find the moment generating function for <math>X_1</math>, <math>S_n</math>, <math>A_n</math>, | ||
Latest revision as of 00:06, 15 June 2024
Let [math]X_1[/math], [math]X_2[/math], ..., [math]X_n[/math] be an independent trials process with density
[[math]]
f(x) = \frac12 e^{-|x|}, \qquad -\infty \lt x \lt +\infty\ .
[[/math]]
- Find the mean and variance of [math]f(x)[/math].
- Find the moment generating function for [math]X_1[/math], [math]S_n[/math], [math]A_n[/math], and [math]S_n^*[/math].
- What can you say about the moment generating function of [math]S_n^*[/math] as [math]n \to \infty[/math]?
- What can you say about the moment generating function of [math]A_n[/math] as [math]n \to \infty[/math]?