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Section 8.5 : Probability

1. Let, \[f\left( x \right) = \left\{ {\begin{array}{ll}{\displaystyle \frac{3}{{37760}}{x^2}\left( {20 - x} \right)}&{{\mbox{if }}2 \le x \le 18}\\0&{{\mbox{otherwise}}}\end{array}} \right.\]
   1. Show that \(f\left( x \right)\) is a probability density function.
   2. Find \(P\left( {X \le 7} \right)\).
   3. Find \(P\left( {X \ge 7} \right)\).
   4. Find \(P\left( {3 \le X \le 14} \right)\).
   5. Determine the mean value of \(X\).
   Solution
2. 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. \[f\left( t \right) = \left\{ {\begin{array}{ll}{0.04{{\bf{e}}^{ - \,\,\frac{t}{{25}}}}}&{{\mbox{if }}t \ge 0}\\0&{{\mbox{if }}t < 0}\end{array}} \right.\]
   1. Verify that \(f\left( t \right)\) is a probability density function.
   2. What is the probability that a light bulb will have a life span less than 8 months?
   3. What is the probability that a light bulb will have a life span more than 20 months?
   4. What is the probability that a light bulb will have a life span between 14 and 30 months?
   5. Determine the mean value of the life span of the light bulbs.
   Solution
3. 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