Chapter 6: Q. 41 (page 378)

Refer to Exercise $37$ . The ferry company’s expenses are $$ 20$ per trip. Define the random variable $Y$ to be the amount of profit (money collected minus expenses) made by the ferry company on a randomly selected trip. That is, $Y = M - 20$ .
(a) How does the mean of $Y$ relate to the mean of $M$ ? Justify your answer. What is the practical importance of $μ_{Y}$ ?
(b) How does the standard deviation of $Y$ relate to the standard deviation of $M$ ? Justify your answer. What is the practical importance of $σ_{Y}$ ?

Short Answer

Expert verified

(a) $μ_{Y} = μ_{M} - 20$

The expected loss per trip is $$ 0.65$ .

(b) $σ_{Y} = σ_{M} = 6.45$

The expected costs are expected to vary by about $$ 6.45$ from the expected loss pf- $$ 0.65$ per trip.

Step by step solution

Part (a) Step 1: Given Information

Given:

$Y = M - 20$

Part (a) Step 2: Explanation

Result of previous exercise:

$μ_{M} = 19.35 σ_{M} = 6.45$

Property mean:

$μ_{a X + b} = a μ_{X} + b$

Then we can determine the mean for $Y$ if we know the mean of $M$ :

$μ_{Y} = μ_{M - 20} = μ_{M} - 20 = 19.35 - 20 = - 0.65$ .

Part (a) Step 1: Given Information

Given:

$Y = M - 20$

Part (b) Step 2: Explanation

Property standard deviation:

$σ_{a X + b} = a σ_{X}$

Result of the previous exercise:

$μ_{M} = 19.35 σ_{M} = 6.45$

Then we can determine the standard deviation for $Y$ :

$σ_{Y} = σ_{M - 20} = σ_{M} = 6.45$

The practical importance is that the fixed costs do not influence the spread of the distribution of money (profit/money collected).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

Part (a) Step 1: Given Information

Part (b) Step 2: Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.