Transcript Working with means and variances…

Working with means
and variances…
Expected Values…

How does an insurance company determine its
premiums?
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Weighs probability of “pay-out” against total revenue
Determines cost per file by calculating yearly operating
costs, amortizations and returns to shareholders
Let’s consider question 4.79 in text and assume
ahead of time that the insurance company must
“make” $300 on this file to meet its financial targets.
Is a premium of $250/a enough?
Life Insurance Premium…
0.9906
Why
“negative”?
Why does the earning
cell become positive
$1250 at age 26?
(-$99,750)x(0.00183) = -$182.54
So, what’s the
expected return?
Mean Value or Expected Value
Value of X
x1
x2
x3
x4
Probability
p1
p2
p3
p4
 X  x1 p1  x2 p2  x3 p3 
 X   xi pi
 xk pk
Rules for Means and Variances

These sound scarier than they really are!

Example: Suppose your mean income is $3200 and
month and your spouse’s income is $2950 per month.
What is your mean monthly family income?
Rule 1:
 X Y   X  Y

You are sending weight data on Canadian grade 7
students to a colleague in Germany. The average
weight is 104 lbs and you need to convert this to kg.
Just before sending the data you discover that your
grad student used a scale that overestimated all of
the weights by 1.2 lbs. What average weight
should you use in your study (expressed in kg)?
(Hint: 1 lb = 0.454 kg)
Rule 2:
a bX  a  b X
Rules for Variances…


Remember – variance is just (standard deviation)2
Variance is just another measure of scatter
Value of X
Probability
x1
p1
x2
p2
 X2  ( x1   X ) 2 p1  ( x2   X ) 2 p2 
  ( xi   X ) pi
2
x3
p3
x4
p4
 ( xk   X ) 2 pk
Rules for handling variances…

If you scale all numbers in a set by a fixed amount
“b” and add a constant amount “a” (ie: a linear
transformation) then:
Rule 3:

2
a bX
b 
2
2
X
Rules for handling variances…

If X and Y are independent random variables,
then:
Rule 4:
  
2
2
2
 X Y   X   Y
2
X Y
2
X
2
Y
Rules for handling variances…

If X and Y are dependent random variables that
have a correlation of r, then:
Rule 4:
      2 r X  Y
2
2
2
 X Y   X   Y  2 r X  Y
2
X Y
2
X
2
Y
Compare Examples 4.25 and 4.26 from text
Why does stats work?

The law of large
numbers…
In conclusion …
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There is a lot stuff in this section! Work
through all of the examples in 4.4
Don’t memorize the formulas – instead try to
“reason” them out in words – what do they
mean
Try the following questions: 4.60, 4.63,
4.67,4.83