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### Section 2-5 : Probability

1. Let, $f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{3}{4}\left( {2x - {x^2}} \right)}&{{\mbox{if }}0 \le x \le 2}\\0&{{\mbox{otherwise}}}\end{array}} \right.$
1. Show that $$f\left( x \right)$$ is a probability density function.
2. Find $$P\left( {X \le 0.25} \right)$$.
3. Find $$P\left( {X \ge 1.4} \right)$$.
4. Find $$P\left( {0.1 \le X \le 1.2} \right)$$.
5. Determine the mean value of $$X$$.
2. Let, $f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{4}{{\ln \left( 3 \right)\left( {4x + {x^2}} \right)}}}&{{\mbox{if 1}} \le x \le 6}\\0&{{\mbox{otherwise}}}\end{array}} \right.$
1. Show that $$f\left( x \right)$$ is a probability density function.
2. Find $$P\left( {X \le 1} \right)$$.
3. Find $$P\left( {X \ge 5} \right)$$.
4. Find $$P\left( {1 \le X \le 5} \right)$$.
5. Determine the mean value of $$X$$.
3. Let, $f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{1}{{10}}\left( {1 + \sin \left( {\pi x - \frac{\pi }{2}} \right)} \right)}&{{\mbox{if 0}} \le x \le 10}\\0&{{\mbox{otherwise}}}\end{array}} \right.$
1. Show that $$f\left( x \right)$$ is a probability density function.
2. Find $$P\left( {X \le 3} \right)$$.
3. Find $$P\left( {X \ge 5} \right)$$.
4. Find $$P\left( {2.5 \le X \le 7} \right)$$.
5. Determine the mean value of $$X$$.
4. The probability density function of the life span of a battery is given by the function below, where t is in years. $f\left( t \right) = \left\{ {\begin{array}{ll}{1.25{{\bf{e}}^{ - \,1.25t}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.$
1. Verify that $$f\left( t \right)$$ is a probability density function.
2. What is the probability that a battery will have a life span less than 10 months?
3. What is the probability that a battery will have a life span more than 2 years?
4. What is the probability that a battery will have a life span between 1.5 and 4 years?
5. Determine the mean value of the life span of the batteries.
5. The probability density function of the successful outcome from some experiment is given by the function below, where t is in minutes. $f\left( t \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{1}{9}\,t\,{{\bf{e}}^{ - \,\,\frac{t}{3}}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.$
1. Verify that $$f\left( t \right)$$ is a probability density function.
2. What is the probability of a successful outcome happening in less than 12 minutes?
3. What is the probability of a successful outcome happening after 25 minutes?
4. What is the probability of a successful outcome happening between 10 and 75 minutes?
5. What is the mean time of a successful outcome from the experiment?
6. Determine the value of $$c$$ for which the function below will be a probability density function. $f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\left( {12{x^4} - {x^5}} \right)}&{{\mbox{if }}0 \le x \le 12}\\0&{{\mbox{otherwise}}}\end{array}} \right.$
7. Use the function below for this problem and assume $$a > 0$$. $f\left( x \right) = \left\{ {\begin{array}{*{20}{l}}{c\,{{\bf{e}}^{ - \,\,\frac{1}{a}\,\,x}}}&{x \ge 0}\\0&{x < 0}\end{array}} \right.$
1. Determine the value of $$c$$ for which this function will be a probability density function.
2. Using the value of $$c$$ found in the first part determine the mean value of the probability density function.