Chapter 6.2, Problem 56E

(a)

To determine

Whether it is possible to calculate the mean of total income X + Y to be ${\mu }_X+{\mu }_Y$ .

Answer to Problem 56E

Yes, we can calculate the mean of total income X + Y to be ${\mu }_X+{\mu }_Y$ .

Explanation of Solution

Given information:

X : Income of husband

Y : Income of wife

We know that

X represents the income of husband.

Y represents the income of wife.

For two random variables,

The mean of sum is equal to their sum of means.

  ${\mu }_{X+Y}={\mu }_X+{\mu }_Y$

As the property holds for any two random variables,

And

Since the means for both X and Y are known,

We can calculate the mean of total income X + Yto be ${\mu }_X+{\mu }_Y$ .

(b)

To determine

Whether it is possible to calculate the variance of total income to be ${\sigma }_X^2+{\sigma }_Y^2$ .

Answer to Problem 56E

No, because X and Y are not independent.

Explanation of Solution

Given information:

X : Income of husband

Y : Income of wife

We know that

X represents the income of husband.

Y represents the income of wife.

For two random variables,

The variance of the sum is equal to the sum of their variances.

However,

It is possible only when two random variables are independent, otherwise this does not hold.

  ${\sigma }_{X+Y}^2={\sigma }_X^2+{\sigma }_Y^2$

Since the intelligent people tend to become more intelligent people.

Also,

The intelligent people tend to earn more money.

Thus,

It is appropriate to assume that the husband’s income X is independent of wife’s income Y .

Hence,

We cannot calculate the variance of total income as ${\sigma }_X^2+{\sigma }_Y^2$ .