General Notice
I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 8.5 : Probability
- Let,
\[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{3}{{37760}}{x^2}\left( {20 - x} \right)}&{{\mbox{if }}2 \le x \le 18}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
- Show that \(f\left( x \right)\) is a probability density function.
- Find \(P\left( {X \le 7} \right)\).
- Find \(P\left( {X \ge 7} \right)\).
- Find \(P\left( {3 \le X \le 14} \right)\).
- Determine the mean value of \(X\).
- For a brand of light bulb the probability density function of the life span of the light bulb is given by the function below, where t is in months.
\[f\left( t \right) = \left\{ {\begin{array}{ll}{0.04{{\bf{e}}^{ - \,\,\frac{t}{{25}}}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.\]
- Verify that \(f\left( t \right)\) is a probability density function.
- What is the probability that a light bulb will have a life span less than 8 months?
- What is the probability that a light bulb will have a life span more than 20 months?
- What is the probability that a light bulb will have a life span between 14 and 30 months?
- Determine the mean value of the life span of the light bulbs.
- Determine the value of \(c\) for which the function below will be a probability density function. \[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\left( {8{x^3} - {x^4}} \right)}&{{\mbox{if }}0 \le x \le 8}\\0&{{\mbox{otherwise}}}\end{array}} \right.\] Solution