General Notice

This is a little bit in advance, but I wanted to let everyone know that my servers will be undergoing some maintenance on May 17 and May 18 during 8:00 AM CST until 2:00 PM CST. Hopefully the only inconvenience will be the occasional “lost/broken” connection that should be fixed by simply reloading the page. Outside of that the maintenance should (fingers crossed) be pretty much “invisible” to everyone.

Paul

May 6, 2021

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*i.e.*you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.Assignment Problems Notice

Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

### Section 2-5 : Probability

- Let,
\[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{3}{4}\left( {2x - {x^2}} \right)}&{{\mbox{if }}0 \le x \le 2}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
- Show that \(f\left( x \right)\) is a probability density function.
- Find \(P\left( {X \le 0.25} \right)\).
- Find \(P\left( {X \ge 1.4} \right)\).
- Find \(P\left( {0.1 \le X \le 1.2} \right)\).
- Determine the mean value of \(X\).

- Let,
\[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{4}{{\ln \left( 3 \right)\left( {4x + {x^2}} \right)}}}&{{\mbox{if 1}} \le x \le 6}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
- Show that \(f\left( x \right)\) is a probability density function.
- Find \(P\left( {X \le 1} \right)\).
- Find \(P\left( {X \ge 5} \right)\).
- Find \(P\left( {1 \le X \le 5} \right)\).
- Determine the mean value of \(X\).

- Let,
\[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{1}{{10}}\left( {1 + \sin \left( {\pi x - \frac{\pi }{2}} \right)} \right)}&{{\mbox{if 0}} \le x \le 10}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
- Show that \(f\left( x \right)\) is a probability density function.
- Find \(P\left( {X \le 3} \right)\).
- Find \(P\left( {X \ge 5} \right)\).
- Find \(P\left( {2.5 \le X \le 7} \right)\).
- Determine the mean value of \(X\).

- The probability density function of the life span of a battery is given by the function below, where t is in years.
\[f\left( t \right) = \left\{ {\begin{array}{ll}{1.25{{\bf{e}}^{ - \,1.25t}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.\]
- Verify that \(f\left( t \right)\) is a probability density function.
- What is the probability that a battery will have a life span less than 10 months?
- What is the probability that a battery will have a life span more than 2 years?
- What is the probability that a battery will have a life span between 1.5 and 4 years?
- Determine the mean value of the life span of the batteries.

- The probability density function of the successful outcome from some experiment is given by the function below, where t is in minutes.
\[f\left( t \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{1}{9}\,t\,{{\bf{e}}^{ - \,\,\frac{t}{3}}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.\]
- Verify that \(f\left( t \right)\) is a probability density function.
- What is the probability of a successful outcome happening in less than 12 minutes?
- What is the probability of a successful outcome happening after 25 minutes?
- What is the probability of a successful outcome happening between 10 and 75 minutes?
- What is the mean time of a successful outcome from the experiment?

- Determine the value of \(c\) for which the function below will be a probability density function. \[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\left( {12{x^4} - {x^5}} \right)}&{{\mbox{if }}0 \le x \le 12}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
- Use the function below for this problem and assume \(a > 0\).
\[f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\,{{\bf{e}}^{ - \,\,\frac{1}{a}\,\,x}}}&{x \ge 0}\\0&{x < 0}\end{array}} \right.\]
- Determine the value of \(c\) for which this function will be a probability density function.
- Using the value of \(c\) found in the first part determine the mean value of the probability density function.