Chapter 3: Survival Distributions and Life Tables
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3.4 Curtate Future Lifetime random variable, K(x).
Symbols:
# the greatest integer function
=round # down to the nearest integer
# the smallest integer function
=gives the smallest integer that is not
less than #
=
=
K(x) = the curtate-future-lifetime of (x)
= the greatest integer in T(x)
= number of future years survived by (x) prior to death
= number of future birthdays (x) will have the
opportunity to celebrate
Notation:
)()( xTxK == the greatest integer in T(x).
Example:
1)
( ) 3.5
T x
=
2)
( ) 3
T x
=
Chapter 3: Survival Distributions and Life Tables
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Example: Suppose a person was born on Jan 1, 1920 and
passed away on Aug 31, 2000. How old was this person at
death?
T(x):
K(x):
NOTE:
)()()( xTxTxT
<
≤
Thus, K(x) = k k ≤ T(x) < k +1, for k = 0, 1, 2, ….
The probability function of K(x):
P(K(x) = k) = P(k ≤ T(x) < k + 1)
= P(k < T(x) ≤ k + 1)
= k+1qx − kqx
= kpx − k+1px
= kpx qx+k
= k|qx
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