91Ó°ÊÓ

A construction project requires your company to make a sealed proposal. You plan to pay another firm $100,000 to complete the job if you get the contract (by having the lowest quote). How much should you bid to maximize your projected profit if you believe the other participating companies' minimum bid (in thousands of dollars) can be represented as the value of a random variable distributed uniformly on (70, 140)

A random variable is a real-valued function defined over the sample space of a random experiment in probability. That is, the random variable's values correlate to the results of the random experiment. Random variables can be discrete or continuous in nature.

The lowest bid's density function is

$f\left(x\right)=\frac{1}{140-70}$

If not, then it is equal to zero. Let's say we invest x thousand dollars in our offer. We will benefit (x-100) thousands of dollars if our bid wins the competition. In that instance, the profit that can be expected is

$E\left(P\right)=(x-100)·\frac{140-x}{70}=\frac{1}{70}\left(240x-x^2-14000\right)$

Let's discover a value for x that maximizes profit. We have that with distinguishing.

$\frac{d}{dx}E\left(P\right)=\frac{1}{70}(240-2x)=0$

This means that x=120. As a result, the highest profit is

${\left.E\left(P\right)\right|}_{x=120}=(120-100)·\frac{140-120}{70}=\frac{40}{7}$

thousands of dollars