Question on ASM manual problem 6.15

[Thomas H]

This question is about problem 6.15 in the ASM study manual.

In the solution the following relation is written:

E[T(0)] = E[min(T(0),40)] + E[T(40)]*40_p_0


I'm having a difficult time seeing how this was obtained. I thought if would be:

E[T(0)] = E[min(T(0),40) + T(40)]

= E[min(T(0),40)] + E[T(40)]

[ctperng]

This question is about problem 6.15 in the ASM study manual.

In the solution the following relation is written:

E[T(0)] = E[min(T(0),40)] + E[T(40)]*40_p_0


I'm having a difficult time seeing how this was obtained. I thought if would be:

E[T(0)] = E[min(T(0),40) + T(40)]

= E[min(T(0),40)] + E[T(40)]

What you suggested really doesn't make sense: what if T(0) < 40? how do you make sense of T(40) in this case? Is it negative? Which density function do you use when you compute the expectation?

To prove the first formula, let the density function for T(0) be f(x). Then the density function for T(40) will be f(x+40)/s(40)
(call this function f_1(x), i.e. f(x+40) = s(40)f_1(x)); here s(x) is the corresponding survival function for f(x). Then

E[T(0)] =int_{0->infinity} xf(x) dx
=int_{0->40} xf(x) dx + int_{40->infinity} xf(x) dx
=int_{0->40} xf(x) dx + int_{0->infinity} (x+40)f(x+40) dx
=int_{0->40} xf(x) dx + int_{0->infinity} 40f(x+40) dx
+ s(40)int_{0->infinity} xf_1(x) dx
But the sum of the first two terms is E[min(T(0),40)] and s(40) = 40_p_0 by definition, so we are done.

ctperng

[Thomas H]

What you suggested really doesn't make sense: what if T(0) < 40? how do you make sense of T(40) in this case? Is it negative? Which density function do you use when you compute the expectation?



I suggested that because I thought that if T(0) < 40 then T(40) = 0.

[ctperng]

Of course you have every right to suggest.

But note that

T(40) = T(0) -40, conditional on T(0) > 40.

When you put them (i.e. T(0) and T(40)) together in the expectation, you know you need to make sense of it.

This may be a good question for you to think about: Let X be a random variable for a sample space S, and Y a random variable of sample space T, what does X + Y mean?

ctperng

[Abraham Weishaus]

ctpergn is correct.

Another way I like putting it is that you can add up unconditional expectations, but you can't mix conditions.

E[T(0)] is the expected amount of time 0 will live. It is the sum of the expected amount of time until age 40 and the expected amount of time after 40, but unconditional!

e_40 is the amount of time 40 will live, given that he already lived to 40. It is not right to add e_40 to E[T(0)] as you suggest, since these expressions are based on 2 different assumptions: e_40 on the assumption that you lived to 40, E[T(0)] on the assumption that you lived to 0.