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 aim is to find the number of gallons \(x\) for which the cumulative distribution function amounts to 0.95. Following the steps and solving for \(x\), the appropriate value in gallons that the store needs to have every day to make sure the probability of exhausting it is 5%.

Step-by-step solution

Setting the Cumulative Distribution Function

In this step, the cumulative distribution function (CDF) has to be set up. The CDF is obtained by integrating the given pdf from the lower limit of 0 to the variable \(x\). The formula is as follows: \(F(x) = \int_0^x f(t) dt\) where \(f(t) = 12 * t * (1000 - t)^2 / 10^12\)

Solving the integration

After setting up the integral, next step is computing the mathematical operation using the given function \(f(t)\), which is integrated with respect to \(t\) from 0 to \(x\). This requires techniques of integration, which can be solved using combination of power rule and substitution method.

Find x where CDF equals 0.95

After the integration is solved, the resulting function should be set to 0.95 (since the probability of exhausting ice cream supply is 0.05 or, correspondingly, the probability of not exhausting is 0.95) and the value of \(x\) is solved. This calculation will provide the amount of ice cream in gallons that the store should have on hand.