exercise:51301a1114: Difference between revisions

Latest revision as of 22:39, 14 June 2024

Assume that the lifetime of a diesel engine part is a random variable [math]X[/math] with density [math]f_X[/math]. When the part wears out, it is replaced by another with the same density. Let [math]N(t)[/math] be the number of parts that are used in time [math]t[/math]. We want to study the random variable [math]N(t)/t[/math]. Since parts are replaced on the average every [math]E(X)[/math] time units, we expect about [math]t/E(X)[/math] parts to be used in time [math]t[/math]. That is, we expect that

[[math]] \lim_{t \to \infty} E \Bigl(\frac {N(t)}t\Bigr) = \frac 1{E(X)}\ . [[/math]]

This result is correct but quite difficult to prove. Write a program that will allow you to specify the density [math]f_X[/math], and the time [math]t[/math], and simulate this experiment to find [math]N(t)/t[/math]. Have your program repeat the experiment 500 times and plot a bar graph for the random outcomes of [math]N(t)/t[/math]. From this data, estimate [math]E(N(t)/t)[/math] and compare this with [math]1/E(X)[/math]. In particular, do this for [math]t = 100[/math] with the following two densities:

[math]f_X = e^{-t}[/math].
[math]f_X = te^{-t}[/math].

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-<div class="d-none"><math>
+Assume that the lifetime of a diesel engine part is a random variable <math>X</math> with density <math>f_X</math>.  When the part wears out, it is replaced by another
-\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> Assume that the lifetime of a diesel engine part is a random
-variable <math>X</math> with density <math>f_X</math>.  When the part wears out, it is replaced by another
 with the same density.  Let <math>N(t)</math> be the number of parts that are used in time
 <math>t</math>.  We want to study the random variable <math>N(t)/t</math>.  Since parts are replaced on the
@@ Line 21: / Line 14: @@
 <math>E(N(t)/t)</math> and compare this with <math>1/E(X)</math>.  In particular, do this for <math>t = 100</math>
 with the following two densities:
-<ul><li> <math>f_X = e^{-t}</math>.
+<ul style="list-style-type:lower-alpha"><li> <math>f_X = e^{-t}</math>.
 </li>
 <li> <math>f_X = te^{-t}</math>.
 </li>
 </ul>