Chapter 6: Q. 55 (page 390)

Imagine that we randomly select a day from the past $10$ years. Let $X$ be the recorded rainfall on this date at the airport in Orlando, Florida, and $Y$ be the recorded rainfall on this date at Disney World just outside Orlando. Suppose that you know the means data-custom-editor="chemistry" $μ_{X} and μ_{Y}$ and the variances data-custom-editor="chemistry" ${σ_{X}}^{2} and {σ_{Y}}^{2}$ of both variables.
a. Can we calculate the mean of the total rainfall $X + Y to be μ_{X} + μ_{Y}$ ? Explain your answer.
b. Can we calculate the variance of the total rainfall to be ${σ_{X}}^{2} + {σ_{Y}}^{2}$ ? Explain your answer

Expert verified

a. Yes, it is reasonable.

b.No, it's not logical.

Step by step solution

Given :

X

: The recorded rainfall at the airport in Orlando

$Y$ : The recorded rainfall at Disney World just outside Orlando

Yes, it is reasonable.

The reason is because the mean has the condition $μ_{X + Y} = μ_{X} + μ_{Y}$ for all conceivable pairs of variables.

As a result, the mean of $X + Y$ is always the same as $μ_{X} + μ_{Y}$ .

Given :

X

: The recorded rainfall at the airport in Orlando

$Y$ : The recorded rainfall at Disney World just outside Orlando

No, it's not logical.

The reason for this is that the variance property applies only when the variables $X and Y$ are independent.

${σ^{2}}_{X + Y} = {σ^{2}}_{X} + {σ^{2}}_{Y}$

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

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.