Transcript Section 7.2 Second Day Rules for Means and Variances

A life
insurance company sells a term insurance
policy to a 21-year-old male that pays $100,000 if
the insured dies within the next 5 years. The
probability that a randomly chosen male will die each
year can be found in mortality tables. The company
collects a premium of $250 each year as payment
for the insurance. The amount X that the company
earns on this policy is $250 per year, less the
$100,000 it must pay if the insured dies. Here is the
distribution of X. Fill in the missing probability in the
table and calculate the mean profit μx.
22
23
24
25
≥26
Age
21
Profit
-$99,750 -$99,500 -$99,250 -$99,000 -$98,750 $1250
Prob.
.00183 .00186 .00189 .00191 .00193
Section 6.2 Second Day
Rules for Means and Variances
Law of Large Numbers


This is very important!
This law says that
As the number of observations increases, the sample mean approaches the idealized mean.
In symbols, as n  , x  

So, the more samples
we get, the closer the
mean is to what is
“should” be.
Rules for Means
If X and Y are random variables, and a and b
are fixed numbers, then
a bX  a  b x
 X Y   X  Y
We will look at these
individually in one
minute…
Simply Put

If you want to find the mean of the sum/difference of
two random variables, you just add/subtract their
means.
–
If the mean of X is 150 and the mean of Y is 1000, then the
mean of X + Y = 150 + 1000.
 x y   x   y
The other rule
a bx  a  b x
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
This says that if we add a number, a, to each sample
then we add a to the mean.
Also, if we multiply every value in the sample by b
then we have to multiply the mean by b.
Rules for Means Demonstrated
X = units sold
for military
division
Probability
1000
3000
5000
10,000
0.1
0.3
0.4
0.2
Y = units sold
for civilian
division
300
500
750
Probability
0.4
0.5
0.1
If this company makes $2000 for each military unit sold and $3500 on
each civilian unit sold, find the mean TOTAL profit.
Rules for Variances
If X is a random variable,
and a and b are fixed
numbers, then …

2
a  bx
b 
2
2
x
If we add the same number,
a, to each item in the
sample, it doesn’t change
the variance. If we multiply
each value by b then the
standard deviation is
multiplied by b, so the
variance is multiplied by the
square of b.
When you have two variables…



We have to look at how the correlation between the
two affect the variance of the sum of x and y
The true correlation is called rho, ρ.
The general rule for variances of random variables…

2
x y
     2  x y

2
x y
     2  x y
2
x
2
x
2
y
2
y
What if they are independent?


If x and y are independent, they have no
effect on one another, so….
ρ = 0 and therefore…

2
x y
2
x
  
2
y

2
x y
  
2
y
2
x
Examples…
If X and Y are independent random
variables and…
 X  5and  Y  2, find  X+Y.
Find  X Y . Find  3 X 4Y .
One more example:
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Suppose Tom’s score in a round of golf is the random
variable X and George’s score in a round of golf is
the random variable Y. Assume their scores are
independent.
Find the average of their combined scores and the
standard deviation of their combined scores if
 X  110; X  10 and Y  100; Y  8
More information…
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One of the skills you need to learn from this
section is combining two independent normal
random variables and finding probabilities.
Find the combined mean and standard deviation,
and then work the problem as you would any
normal curve probability (find the Z-score).
In a fair game,   0
Example
Go back to Tom and George’s golf game. What
percent of the time would we expect Tom to win?
 Tom:  = 110 =10
 George:  = 100  = 8
  (x-y) = 10
 (x-y) = 12.8
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Homework
Chapter 6
#37, 38, 42, 44, 47, 48,
56