Math  /  Calculus

Question1. The life expectancy in a demographic model is a random variable with a distribution given by the density g(t)=μ1e100μeμt1[0,100](t)g(t)=\frac{\mu}{1-e^{-100 \mu}} e^{-\mu t} 1_{[0,100]}(t) for some parameter μ>0\mu>0. Determine the median and the mean life expectancy in this model.

Studdy Solution
Solve for the mean:
Integrate by parts, let u=t u = t and dv=μ1e100μeμtdt dv = \frac{\mu}{1-e^{-100 \mu}} e^{-\mu t} \, dt .
du=dt du = dt v=[11e100μeμt] v = \left[ -\frac{1}{1-e^{-100 \mu}} e^{-\mu t} \right]
Apply integration by parts:
E[T]=[t1e100μeμt]0100+010011e100μeμtdt \mathbb{E}[T] = \left[ -\frac{t}{1-e^{-100 \mu}} e^{-\mu t} \right]_0^{100} + \int_0^{100} \frac{1}{1-e^{-100 \mu}} e^{-\mu t} \, dt
Calculate:
E[T]=(1001e100μe100μ+0)+11e100μ[1μeμt]0100 \mathbb{E}[T] = \left( -\frac{100}{1-e^{-100 \mu}} e^{-100 \mu} + 0 \right) + \frac{1}{1-e^{-100 \mu}} \left[ -\frac{1}{\mu} e^{-\mu t} \right]_0^{100}
E[T]=100e100μ1e100μ+1μ(1e100μ)(1e100μ) \mathbb{E}[T] = -\frac{100 e^{-100 \mu}}{1-e^{-100 \mu}} + \frac{1}{\mu(1-e^{-100 \mu})} (1 - e^{-100 \mu})
E[T]=1μ100e100μ1e100μ \mathbb{E}[T] = \frac{1}{\mu} - \frac{100 e^{-100 \mu}}{1-e^{-100 \mu}}
The median life expectancy is:
m=1μln(0.5+0.5e100μ) m = -\frac{1}{\mu} \ln(0.5 + 0.5e^{-100 \mu})
The mean life expectancy is:
E[T]=1μ100e100μ1e100μ \mathbb{E}[T] = \frac{1}{\mu} - \frac{100 e^{-100 \mu}}{1-e^{-100 \mu}}

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