1. Random Variables
Chapter 16

2. Expected Value: Center
A random variable assumes a value based on the outcome of a random event.
We use a capital letter, like X, to denote a random variable.
A particular value of a random variable will be denoted with a lower case letter, in this case x.

3. Expected Value: Center (cont.)
There are two types of random variables:
Discrete random variables can take one of a finite number of distinct outcomes.
Example: Number of credit hours
Continuous random variables can take any numeric value within a range of values.
Example: Cost of books this term

4. Expected Value: Center (cont.)
A probability model for a random variable consists of:
The collection of all possible values of a random variable, and
the probabilities that the values occur.
Of particular interest is the value we expect a random variable to take on, notated μ (for population mean) or E(X) for expected value.

5. Example
You pick one card from a deck of cards. If you pick a face card you win $15. If you pick an ace you win $25, and if you pick the ace of hearts you win $65. For any other card you win nothing.
Create a probability model for the amount you win at this game.
Amount won $0
$15
$25
$65
P(Amount won)
36/52
12/52
3/52
1/52

6. Expected Value: Center (cont.)
The expected value of a (discrete) random variable can be found by summing the products of each possible value by the probability that it occurs:
Note: Be sure that every possible outcome is included in the sum and verify that you have a valid probability model to start with.
Find the expected value of the card example

7. First Center, Now Spread…
For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with random variables as well.
The variance for a random variable is:
The standard deviation for a random variable is:

8. More About Means and Variances
Adding or subtracting a constant from data shifts the mean but doesn’t change the variance or standard deviation:
E(X ± c) = E(X) ± c Var(X ± c) = Var(X)
Example: What happens if we add $100 to each of the winning amounts of the card game.

9. More About Means and Variances (cont.)
In general, multiplying each value of a random variable by a constant multiplies the mean by that constant and the variance by the square of the constant:
E(aX) = aE(X) Var(aX) = a2Var(X)
Example: Consider everyone in a company receiving a 10% increase in salary.

10. More About Means and Variances (cont.)
In general,
The mean of the sum of two random variables is the sum of the means.
The mean of the difference of two random variables is the difference of the means.
E(X ± Y) = E(X) ± E(Y)
If the random variables are independent, the variance of their sum or difference is always the sum of the variances.
Var(X ± Y) = Var(X) + Var(Y)

11. example
Miguel buys a large bottle and a small bottle of juice. The amount of juice that the manufacturer puts in the large bottle is a random variable with a mean of 1024 ml and a standard deviation of 12 ml. The amount of juice that the manufacturer puts in the small bottle is a random variable with a mean of 508 ml and a standard deviation of 4 ml. Find the mean and standard deviation of the total amount of juice in the two bottles.

12. Example
Suppose that in one town, 50 year old men have a mean weight of 177 lb. with a standard deviation of 17 lb. 30 year old men have a mean weight of 158 lb. with a standard deviation of 12 lb. How much heavier do you expect a 50 year old man to be than a 30 year old man and what is the standard deviation of this difference?

13. Example
A national study found that the average family spent $422 a month on groceries, with a standard deviation of $84. The average amount spent on housing (rent or mortgage) was $1120 a month, with standard deviation $212. What is mean and standard deviation of the total?

14. Example Runner 1 2 3 4 Total Mean 12.17 12.24 11.94 11.79 SD 0.10 0.15
In the 4 × 100 relay event, each of four runners runs 100 meters. A college team is preparing for a competition. The means and standard deviations of the times (in seconds) of their four runners are as shown in the table What are the mean and standard deviation of the relay team's total time in this event? Assume that the runners' performances are independent.
Runner 1 2 3 4
Total
Mean
12.17
12.24
11.94
11.79 SD
0.10
0.15
0.08

15. Continuous Random Variables (cont.)
Good news: nearly everything we’ve said about how discrete random variables behave is true of continuous random variables, as well.
When two independent continuous random variables have Normal models, so does their sum or difference.
This fact will let us apply our knowledge of Normal probabilities to questions about the sum or difference of independent random variables.

16. Example
Sue buys a large packet of rice. The amount of rice that the manufacturer puts in the packet is a random variable with a mean of 1016 g and a standard deviation of 8 g. The amount of rice that Sue uses in a week has a mean of 210 g and a standard deviation of 40 g. If the weight of the rice remaining in the packet after a week can be described by a normal model, what's the probability that the packet still contains more than g after a week?