Survival Function (SF) Method in finding the Expected Value (aka Darth
Vader rule)
I hope this will be a sufficiently comprehensive explanation on the survival
function method in finding the expected value. However here are some
comments:
 I personally DO NOT find the SF method that great due to its
restrictions etc, except when you are dealing with an exponential
distribution (avoids integration by parts). Even though tabular
integration is great, it almost seems like the SF method is “made for”
cases with an exponential distribution.
 All formulas given here would be in the continuous case, because this
method is most commonly used when the random variable is
continuous. The SF method can be used for discrete cases, but it can
get a little crazy and won’t be covered here.
 I DON’T think it is worthwhile putting as much effort to study and
memorize the SF method as compared to other important concepts.
So, this is largely for those who would like to “know everything” for
the fun of it. (yes, I confess…)
 All the formulas, subject to its case, are in red, so if you don’t care to
know why, just look at those.
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Expected Value of
Condition: Random variable
When
must be non-negative, ie.
.
,
( )
∫
( )
Think of how you would describe this formula in words. The expected value
of is the “sum” of the SFs of from 0 to . I’ll borrow the discrete case to
illustrate. Remember that ( ) is the sum of times the probability of ,
and can be written as:
( )
(
(
)
)
(
)
The table below splits these terms up and arranges them in rows. The sum
of each row is in the right-most column. Notice that the column consists of
the SFs evaluated at an integer beginning at 0.
(
)
(
)
(
(
)
)
(
(
(
(
)
)
)
)
(
(
(
(
( )
)
So it’s no surprise that by summing the SFs from 0 to
In general, for ( ) where ( )
[ ( )]
)
)
)
, we get back ( ).
,
∫
( )
( )
This can be derived using integration by parts. This formula can be used to
find the moments of X.
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But when
,
( )
∫
( )
This is actually using the very first formula, keeping in mind that between
0 and 2, ( )
and between 5 and , ( )
. In general:
When
such that
,
( )
∫
∫
∫
( )
( )
∫
∫
( )
If you like, you could reason why this is true by assuming the discrete case
and using the table method above. You might appear to run into some
contradictions, but keep in mind that
(
),
(
)
) too] are 0 as a result of
and (
) [and therefore (
assuming a discrete setup. Also, remember that (
) is 1. One more
thing: be careful not to interchange the discrete and continuous cases too
loosely.
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Expected Value of Payment ( )
You could either find ( ) and use the SF method directly, or use the SF
method on ( ) instead with some modifications. The latter usage will be
covered here.
There are 3 typical cases when there is an insurance payment
amount :
on a loss
Deductible only:
{
( )
( )
∫
This can be proven with the integration by parts technique similar to finding
[ ( )]. We start with the original formula with a twist:
( )
∫ (
( )
( )
)
( )] |
∫
(
∫
( )
And just like how finding [ ( )] requires
)
∫
( )
]
)
( )
∫
(
( ))
;
;
Let
[[(
)(
(
( ):
)
( )
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Policy limit only:
{
( )
( )
∫
Proof:
( )
∫
( ))
(
Let
[[
( )
( )
;
;
( )] |
( )
(
∫
∫
∫
( )
( )
( )
]
)
( )
( )
( )
And again:
(
)
∫
( )
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Deductible and policy limit:
Here,
is defined as the maximum payment by the insurer.
{
( )
( )
∫
Proof:
( )
(
∫
)(
Let
[[(
)
(
( ))
(
( )
∫
)
)
( )
( )
;
;
( )] |
(
( )
∫
]
)
(
(
)
)
( )
∫
And again:
(
)
∫
(
)
( )
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However, those were all cases where
. What if
, where
? As it turns out, only the “policy limit only” case is affected. This
should be quite intuitive, since it is the only case that relies on 0 as the lower
limit.
When
, such that
,
Policy limit only:
( )
(
)
∫
∫
( )
( )
Written by JLWJ from Coaching Actuaries. Please visit coachingactuaries.com