Q.5.5 - A First Course in Probability

Question

 A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function

$f\left(x\right) = \left\{\begin{array}{ll}5{(1 - x)}^4& 0 < x < 1\\ 0& otherwise\end{array}\right.$

what must the capacity of the tank be so that the probability of the supply being exhausted in a given week is $.01?$

Step-by-Step Solution

Answer

 the capacity of the tank $c=0.602$ 

Step 1 Given Information.

$f\left(x\right) =\left\{\begin{array}{ll} 5{(1 - x)}^4& 0 < x < 1\\ 0& otherwise\end{array}\right.$

Step 2 Explanation.

$c\Rightarrow p(x>c) =0.07$

${\int }_c^{1}5{(1-x)}^{4}dx=0.01$

${{\left[\frac{-5{(1-x)}^5}{5}\right]}_c}^1=0.07$

Step 3 Explanation.

$1-c ={(0.01)}^{\frac{1}{5}}$

$c={(1-0.01)}^{\frac{1}{5}}$

$=0.602$