Calculating Life expectancy

Stuck with the following problem:

The Probability that a newborn will die at or before time t is given by the expression:

F₀

(t)=1-(1-{t/100})^{1/4}

0\le{t}\le{100}

Calculate the life expectancy of a person aged 40. How I've started:

Let T_x be the time of death of a person aged x.

Pr(T

\le{t}) = F

(t)

Let

Pr(T

>{t}) = S

(t)=1-F

(t)

Now Life expectancy (Expected number of years that x will live)

E(T

) = \int(tf

(t)) dt

where

f

(t)

is the density function of F.

Integration by parts gives:

E(T

) = \int(S

(t)) dt

To find

S

(t)

Pr(T

>(t)) = Pr(T

>x+t\Bigm|T

>x)

Using the law of conditional Probability:

Pr(T

>(t)) = (Pr(T

>x+t))/(Pr(T

>x))

Pr(T

>(t)) = S

(x+t)/S

(x)

S

(x) = (1-{x/100})^{1/4}

So now to calculate Life expectancy,

E(T

) = \int (1+({t/({x-100)}}))^{1/4}dt

In our case, x = 40 and the integral is defined from 0 to 100, so

E(T

₄₀

) = \int_0^{100} (1+({t/-60}))^{1/4}dt

I've integrated this to get:

{4/5}(x-100)((t+x-100)/(x-100))^{5/4}

However when subbing in 100 to find the definite integral, you end up having to calculate (-2/3)^({5/4}). What do I do?

(edited 10 years ago)

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