Question

Let \(X\) be the number of gallons of ice cream that is requested at a certain store on a hot summer day. Assume that \(f(x)=12 x(1000-x)^{2} / 10^{12}, 0

Short answer

The exact value of \( x \) will need to be calculated using numerical methods or suitable software, as the equation derived in Step 3 might not have a simple analytical solution. This will give the number of gallons the store should stock each day to have a 5% chance of running out.

Step-by-step solution

Finding the cumulative distribution function (CDF)

Integrate the given probability density function \( f(x)=\frac{12x(1000-x)^{2}}{10^{12}} \) with limits from 0 to \( x \) to find CDF. The CDF \( F(x) = \int_{0}^{x}f(t)dt \)

Equal the CDF to 0.95

To find the amount of ice cream that the store should keep so that there is 95% chance that this quantity will be sufficient, solve \( F(x) = 0.95 \)

Solve the equation for \( x \)

Solving the equation \( F(x) = 0.95 \) will give us the value of \( x \), which represents the number of gallons of ice cream the store should have on hand each day in order to maintain a 5% chance of exhausting its supply.