Chapter 6.2, Problem 73E

To determine

Mean for the random variable representing the total number of the calories in the breakfast.

Answer to Problem 73E

The mean of T is 180.

Explanation of Solution

Given information:

For 1 − ounce breakfast cereal:

Mean, ${\mu }_X=110$

Variance, ${\sigma }_X=10$

For cup of whole milk:

Mean, ${\mu }_Y=140$

Variance, ${\sigma }_Y=12$

Calculations:

If X and Y are independent,

Property mean:

  ${\mu }_{aX+bY}=a{\mu }_X+b{\mu }_Y$

Property variance:

  ${\sigma }_{aX+bY}^2=a^2{\mu }_X^2+b^2{\mu }_Y^2$

Since T represents the total number of calories in breakfast (1 ounce of the cereal and half cup of whole milk).

Then

  $T=X+Y/2$

Thus,

We have

Mean of T ,

  ${\mu }_T={\mu }_{X+Y/2}={\mu }_X+\frac{1}{2}{\mu }_Y=110+\frac{1}{2}\left(140\right)=180$

Variance of T ,

  ${\sigma }_T={\sigma }_{X+Y/2}=\sqrt{{\sigma }_X^2+\frac{1}{2^2}{\sigma }_Y^2}=\sqrt{{\left(10\right)}^2+\frac{1}{4}{\left(12\right)}^2}\approx 11.66$