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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 5: Q. 5.6 (page 217)

Your company must make a sealed bid for a construction project. If you succeed in winning the contract (by having the lowest bid), then you plan to pay another firm $$ 100, 000$ to do the work. If you believe that the minimum bid (in thousands of dollars) of the other participating companies can be modeled as the value of a random variable that is uniformly distributed on $(70, 140)$ , how much should you bid to maximize your expected profit?

Short Answer

Expert verified

The maximum profit to be expected is $40 / 7$ thousands of dollars

Step by step solution

01

Evaluate E(P)

The lowest bid's density function is

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

Otherwise, it is equal to zero for $x \in (70, 140)$ . Let's say we invest $x$ thousand dollars in our offer. We will make a profit of $(x - 10)$ thousand dollars if our bid wins the competition. In that instance, the profit that can be expected is

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

02

Find Maximum profit

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

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

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

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

thousand of dollars.

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Most popular questions from this chapter

Repeat Problem $6.2$ when the ball selected is replaced in the urn before the next selection.

If $x$ is a beta random variable with parameters $a$ and $b$ , show that

$E [X] = \frac{a}{a + b}$

$Var (X) = \frac{ab}{(a + b)^{2} (a + b + 1)}$

The annual rainfall (in inches) in a certain region is normally distributed with $μ = 40 and σ = 4$ . What is the probability that starting with this year, it will take more than $10$ years before a year occurs having a rainfall of more than $50$ inches? What assumptions are you making?

Define a collection of events $E_{a}, 0 < a < 1$ having the property that $P (E_{a}) = 1$ for all $a$ but $P (\underset{a}{\cap} E_{a}) = 0$

Hint: Let $X$ be uniform over $(0, 1)$ and define each $E_{a}$ in terms of $X$ .

If has a hazard rate function $λ_{X} (t)$ , compute the hazard rate function of $a X$ where $a$ is a positive constant.

See all solutions

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