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### Section 2-5 : Probability

1. Let, $f\left( x \right) = \left\{ {\begin{array}{ll}\frac{3}{{37760}}{x^2}\left( {20 - x} \right)}&{{\mbox{if }}2 \le x \le 18}\\0&{{\mbox{otherwise}}}\end{array}} \right$
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