1. More on
Random
Variables

2. Suppose that X and Y are random variables with μx = 25, σ²x = 3, μy = 30, and σ²y = 4. What are the expected value and variance of the random variable (X + Y)?
A) μ(x + y) = 55, σ²(x + y) = 3.5
B) μ(x + y) = 55, σ²(x + y) = 5
C) μ(x + y) = 55, σ²(x + y) = 7
D) μ(x + y) = 27.5, σ²(x + y) = 7
E) There is insufficient information to answer this question

3. The Law of Large Numbers
We can decide how accurately we want to estimate the mean of a population.
We can draw a number of independent observations from a any population with a finite mean = µ.
As the number of observations increases, the mean of those observations eventually approaches µ as closely as we specified, and it remains that close to µ.

4. Average results of random independent trials are predictable and stable.
They are predictable because any random phenomenon is predictable over time.
They are stable because as the sample size increases, each subsequent observation has a smaller effect on the sample mean.

5. You are studying flaws in the painted finish of refrigerators
You are studying flaws in the painted finish of refrigerators. There are two types of flaws you are looking for – dimples and sags. Obviously not every refrigerator has the same number of flaws.
What is the average number of total flaws per refrigerator if the average number of dimples is 0.7 and the average number of sags is 1.4?
= 2.1

6. Rules for Means of Random Variables
µ(X + Y) = µX + µY
μ(X – Y) = μX – μY

7. Find the average length of these roaches.
As part of a biology assignment, a college student wants to estimate the average length of the roaches in his dorm. He collects ten roaches, and finds their lengths, in inches, are as follows:
Find the average length of these roaches.
μ(roach) = 1.09 inches

8. Unfortunately for this student, his professor wants all measurements in metric units. So the roaches should have been measured in centimeters instead of inches.
Using a conversion factor of 1 inch = 2.54 cm, find the new average length of the roaches.
μ(roach) = cm

9. Rules for Means of Random Variables
µ(X + Y) = µX + µY
μ(X – Y) = μX – μY
μ(bX) = b · μX

10. Gain Communication military sales:
Units sold ,000
Probability
Gain Communication civilian sales:
Units sold
Probability
If Gain makes $2000 profit on every military unit sold, and $3500 profit on every civilian unit sold, calculate the expected profit based on these sales estimates.

11. The mean of the sum of two random variables is the sum of the means of the two random variables.
This approach does not necessarily work when dealing with variances or standard deviations of random variables.

12. X = % of a family’s income saved per month
Y = % of a family’s income spent per month
X and Y will probably vary from month to month.
Regardless of the variation, the sum of X and Y
will always be100%.
X and Y are not independent of each other.

13. Rules for adding variances of two independent random variables:
In order to add the variances of two random variables, we must know that the variables are independent of each other.
Rules for adding variances of two independent random variables:
σ²(X +Y) = σ²X + σ²Y
σ²(X – Y) = σ²X + σ²Y

14. Notice that the standard deviations do not add.
Let’s calculate the mean, variance, and standard deviation of the expected winnings for a single pick three lottery ticket.
μ = σ² = σ = 15.80
Drawings on consecutive days are independent of each other. Let’s find the mean, variance, and standard deviation of the expected winnings for two lottery tickets played on consecutive days.
Notice that the standard deviations do not add.

15. Suppose that X and Y are random variables with μx = 25, σ²x = 3, μy = 30, and σ²y = 4. What are the expected value and variance of the random variable (X + Y)?
A) μ(x + y) = 55, σ²(x + y) = 3.5
B) μ(x + y) = 55, σ²(x + y) = 5
C) μ(x + y) = 55, σ²(x + y) = 7
D) μ(x + y) = 27.5, σ²(x + y) = 7
E) There is insufficient information to answer this question

16. HOMEWORK:
pages 425 – 426
#36, 37a, 37b, 39
page 429
#49
Test Thursday