1. Chapter 16 Random Variables
math2200

2. Life insurance Insurance company: a “death and disability” policy
Pay $10,000 when the client dies
Pay $5,000 if the client is permanently disabled
Charge $50 per year
Why $50?
Using actuarial information, the company can calculate the expected value of the policy.

3. Random variable
The amount the company pays out on an individual policy is called a random variable.
A random variable assumes a value based on the outcome of a random event.
Random variable is often denoted by a capital letter, e.g., X
A particular value that it can have is often denoted by the corresponding lower case letter, e.g., x

4. Random variable Discrete Continuous
We can list all the outcomes (finite or countable)
E.g. the amount the insurance pays out is either $10,000, $5,000 or $0
Continuous
any numeric value within a range of values.
Example: the time you spend from home to school

5. Probability model
The collection of all possible values and the probabilities that they occur is called the probability model for the random variable.

6. Example Death rate in any year is 1 out of every 1000 people
2 out of 1000 suffer some kind of disability
Probability model
Policyholder outcome
Payment
(x)
Probability (Pr(X=x))
Death
10,000
1/1000
Disability
5,000
2/1000
Neither
997/1000

7. What does the insurance company expect?
Suppose it insures exactly 1000 people
In a year,
1 customer dies
2 are disabled
The insurance company pays $10,000 + $5,000*2 = $20,000
Payment per customer: $20,000/1000 = $20
Earnings per customer: $50
Profit : $30 per customer!

8. Expected value $20 is the expected payment per customer E(X) = 20
=(10000 * * 2 + 0*997) / 1000
=10000*(1/1000) *(2/1000) + 0*(997/1000)
E(X) = Σx* P(X=x)
Center of the distribution
A parameter of the model

9. Expected value
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.
The expected value of a (discrete) random variable can be found by summing the products of each possible value and 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.

10. Most of the time, the company makes $50 per customer
But, with small probabilities, the company needs to pay a lot ($10000 or $5000)
The variation is big
How to measure the variation?

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

12. Variance and standard deviation
Policyholder outcome
Payment (x)
Probability Pr(X=x)
Deviation
Death
10,000
1/1000
( ) = 9980
Disability
5,000
2/1000
=4980
Neither
997/1000
0 -20 = -20
Var(X) = Σ[x-E(X)]2 * P(X=x)
Variance = (1/1000) (2/1000)+(-20)2 (997/1000) = 149,600
Standard deviation = square root of variance
SD(X) = $386.78

13. Properties of expected value and standard deviation
Shifting
E(X+c) = E(X) + c
Var(X+c) = Var(X)
Example: Consider everyone in a company receiving a $5000 increase in salary.
Scaling
E(aX) = aE(X)
Var(aX) = a2 Var(X)
Example: Consider everyone in a company receiving a 10% increase in salary.

14. Properties of expected value and standard deviation
Additivity
E(X ± Y) = E(X) ± E(Y)
If X and Y are independent
Var(X ± Y) = Var(X) + Var(Y)
SD(X+Y) is NOT SD(X)+SD(Y)
Suppose the outcomes for two customers are independent, what is the variance for the total payment to these two customers?
Var(X+Y) = Var(X)+Var(Y) = =
If one customer is insured twice as much, the variance is
Var(2X) = 4Var(X) = 4* =
SD(2X) = 2SD(X)

15. X+Y and 2X Random variables do not simply add up together!
X and Y have the same probability model
But they are not the same random variables
Can NOT be written as X + X

16. Example Sell used Isuzu Trooper and purchase a new Honda motor scooter
Selling Isuzu for a mean of $6940 with a standard deviation $250
Purchase a new scooter for a mean of $1413 with a standard deviation $11
How much money do I expect to have after the transaction? What is the standard deviation?

17. Combining Random Variables (The Bad News)
It would be nice if we could go directly from models of each random variable to a model for their sum.
But, the probability model for the sum of two random variables is not necessarily the same as the model we started with even when the variables are independent.
Thus, even though expected values may add, the probability model itself is different.

18. Combining Random Variables (The Good News)
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.

19. Combining normal random variables
Example: packaging stereos
Stage 1: packing
Normal with mean 9min and sd 1.5min
Stage 2: boxing
Normal with mean 6min and sd 1min
What is the probability that packing two consecutive systems take over 20 minutes?

20. X1: time for packing the first system
mean=9, sd = 1.5
X2: time for packing the second system
T=X1+X2: total time to pack two systems
E(T) = E(X1)+E(X2) = 9+9=18
Var(T) = Var(X1)+Var(X2) = (assuming independence)
T is Normal with mean 18 and sd 2.12
z-score = (20-18)/2.12 = 0.94
P(T>20) = P(Z>0.94) =

21. What percentage of the stereo systems take longer to pack than to box?
P: time for packing a system
B: time for boxing a system
D=P-B: difference in times to pack and box a system
The questions is P(D>0)=?
Assuming P and B are independent

22. D is still Normal
E(D) = E(P-B) = E(P)-E(B) = 9-6=3
Var(D) = Var(P-B) = Var(P)+Var(B) = = 3.25
SD(D) = 1.80
D is Normal with mean 3 and sd 1.80
P(D>0) =
About 95% of all the stereo systems will require more time for packing than for boxing

23. Correlation and Covariance
If X is a random variable with expected value E(X)=µ and Y is a random variable with expected value E(Y)=ν, then the covariance of X and Y is defined as
Cov(X,Y)=E((X-µ)(Y- ν))
The covariance measures how X and Y vary together.

24. Some properties of covariance
Cov(X,Y)=Cov(Y,X)
Cov(X,X)=Var(X)
Cov(cX,dY)=c*dCov(X,Y)
Cov(X,Y) = E(XY)- µν
If X and Y are independent, Cov(X,Y)=0
The converse is NOT true
Var(X ± Y) = Var(X) + Var(Y) ± 2Cov(X,Y)

25. Correlation and Covariance (cont.)
Covariance, unlike correlation, doesn’t have to be between -1 and 1.
To fix the “problem” we can divide the covariance by each of the standard deviations to get the correlation:

26. What Can Go Wrong? Don’t assume everything’s Normal.
You must Think about whether the Normality Assumption is justified.
Watch out for variables that aren’t independent:
You can add expected values for any two random variables, but
you can only add variances of independent random variables.

27. What Can Go Wrong? (cont.)
Don’t forget: Variances of independent random variables add. Standard deviations don’t.
Don’t forget: Variances of independent random variables add, even when you’re looking at the difference between them.
Don’t write independent instances of a random variable with notation that looks like they are the same variables.