exercise:C5e148cb87: 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> Show that if <math>X</math> is a random variable with mean <math>\mu</math> and variance <math>\sigma^2</math>, and if <math>X^* = (X - \mu)/\sigma</math> is the standardized version of <math>X</math>, then <math display="block"> g_{X^*}(t) = e^{...") |
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Show that if <math>X</math> is a random variable with mean <math>\mu</math> and variance <math>\sigma^2</math>, and if <math>X^* = (X - \mu)/\sigma</math> is the standardized version of <math>X</math>, then | |||
and variance <math>\sigma^2</math>, and if <math>X^* = (X - \mu)/\sigma</math> is the standardized | |||
version of <math>X</math>, then | |||
<math display="block"> | <math display="block"> | ||
g_{X^*}(t) = e^{-\mu t/\sigma} g_X\left( \frac t\sigma \right)\ . | g_{X^*}(t) = e^{-\mu t/\sigma} g_X\left( \frac t\sigma \right)\ . | ||
</math> | </math> | ||
Latest revision as of 23:46, 14 June 2024
Show that if [math]X[/math] is a random variable with mean [math]\mu[/math] and variance [math]\sigma^2[/math], and if [math]X^* = (X - \mu)/\sigma[/math] is the standardized version of [math]X[/math], then
[[math]]
g_{X^*}(t) = e^{-\mu t/\sigma} g_X\left( \frac t\sigma \right)\ .
[[/math]]