Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
Title: multiple lives 2
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 2, there are part 1 and part 3.
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 2, there are part 1 and part 3.
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
Multiple Lives
Contingent Survival Functions
1
...
2
𝑡 𝑞𝑥𝑦 : the probability that (x) dies second and within t
...
Assuming states:
- 0: both alive
- 1: only (x) alive
- 2: only (y) alive
- 3: neither alive
2
...
No two lives die simultaneously
...
Two lives are independent
...
Relationships (even without independence)
1
1
∞𝑞𝑥𝑦 + ∞𝑞𝑥𝑦 = 1
2
2
∞𝑞𝑥𝑦 + ∞𝑞𝑥𝑦 = 1
1
2
∞𝑞𝑥𝑦 = ∞𝑞𝑥𝑦
1
1
𝑡 𝑞𝑥𝑦 + 𝑡 𝑞𝑥𝑦 = 𝑡 𝑞𝑥𝑦
2
2
𝑡 𝑞𝑥𝑦 + 𝑡 𝑞𝑥𝑦 = 𝑡 𝑞𝑥𝑦
1
2
𝑛 𝑞𝑥𝑦 = 𝑛𝑞𝑥𝑦 + 𝑛𝑞𝑥 𝑛𝑝𝑦
1
2
𝑛𝑞𝑥 = 𝑛𝑞𝑥𝑦 + 𝑛 𝑞𝑥𝑦
1
1
If 𝜇𝑦+𝑡 = 𝑘𝜇𝑥+𝑡 , then 𝑛𝑞𝑥𝑦
= 1+𝑘 𝑛 𝑞𝑥𝑦
1
If both forces of mortality are constant, then 𝑛𝑞𝑥𝑦
=
𝜇𝑥 (1−𝑒 −𝑛(𝜇𝑥 +𝜇𝑦) )
𝜇𝑥 +𝜇𝑦
Let the two lives (x) and (y) have parameters 𝜔𝑥 and 𝜔𝑦 , 𝑎 = 𝜔𝑥 − 𝑥, 𝑏 = 𝜔𝑦 − 𝑦, then,
𝑛
𝑛2
−
𝑛 ≤ min (𝑎, 𝑏)
𝑎 2𝑎𝑏
𝑎
1
𝑛 ≥ 𝑎 𝑎𝑛𝑑 𝑏 ≥ 𝑎
𝑛𝑞𝑥𝑦 = 1 −
2𝑏
𝑏
𝑛 ≥ 𝑎 𝑎𝑛𝑑 𝑏 ≤ 𝑎
{ 2𝑎
Assuming independence:
2
1
𝑡 𝑞𝑥𝑦 = 𝑡 𝑞𝑥𝑦 − 𝑡 𝑞𝑦 𝑡 𝑝𝑥
𝑡
1
𝑡 𝑞𝑥𝑦 = ∫
𝑠𝑝𝑦 𝑠 𝑝𝑥 𝜇𝑥+𝑠 𝑑𝑠
0
𝑡
2
𝑡 𝑞𝑥𝑦
=∫
𝑠𝑞𝑦 𝑠 𝑝𝑥 𝜇𝑥+𝑠 𝑑𝑠
0
Moments
1
...
For the expected future lifetime of the last survivor status (the expected amount of time to the
second death), we can use 𝑒𝑥𝑦 = 𝑒𝑥 + 𝑒𝑦 − 𝑒𝑥𝑦 (dependent or independent)
...
Two special cases: exponential lifetimes and uniform survival
a
...
For a beta distribution with 𝛼 and 𝜔,
𝜔−𝑥
expected future lifetime for (x) is 𝑒𝑥 = 𝛼+1
o If future survival for each of two lives (x) and (y) is uniform with limiting ages 𝜔𝑥 and
𝜔𝑦 , but 𝜔𝑥 − 𝑥 ≠ 𝜔𝑦 − 𝑦
...
Then,
𝑒𝑥𝑦 = 𝐸[𝑇𝑥𝑦 ] = 𝑎𝑝𝑦 𝐸[𝑇𝑥𝑦 |𝑇𝑌 > 𝑎] + 𝑎𝑞𝑦 𝐸[𝑇𝑥𝑦 |𝑇𝑌 ≤ 𝑎]
o If (y) survives to time a, 𝑇𝑥𝑦 depends only on the lifetime of (x) and is uniform on (0,a],
𝑎
so 𝑇𝑥𝑦 ’s expected value is the midpoint of (0,a], 2
...
Then,
𝑎
𝑎
𝑒𝑥𝑦 = 𝑎𝑝𝑦 ( ) + 𝑎 𝑞𝑦 ( )
2
3
𝑎
Since 𝑎𝑞𝑦 = 𝑏 , the final formula is
𝑎 𝑎
𝑎 𝑎
𝑎 𝑎2 𝑎2 𝑎 𝑎2
𝑒𝑥𝑦 = (1 − ) ( ) + ( ) ( ) = −
+
= −
𝑏 2
𝑏 3
2 2𝑏 3𝑏 2 6𝑏
𝑎 𝑏 𝑎 𝑎2 𝑏 𝑎2
𝑒𝑥𝑦 = 𝑒𝑥 + 𝑒𝑦 − 𝑒𝑥𝑦 = + − +
= +
2 2 2 6𝑏 2 6𝑏
b
...
To calculate the variance of future lifetime for joint life and last survivor statuses, we can use
∞
2
𝑉𝑎𝑟(𝑇𝑥𝑦 ) = 2 ∫ 𝑡 𝑡𝑝𝑥𝑦 𝑑𝑡 − 𝑒𝑥𝑦
0
5
Title: multiple lives 2
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 2, there are part 1 and part 3.
Description: This note allows master's students to learn multiple lives. It is as straightforward as possible. This is part 2, there are part 1 and part 3.