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I am playing around with the mean expected lifetime formula using an exponential distribution. The following is the formula I derived:

$$\frac{{e^{\lambda t_a}}(1+λt_a)-{e^{\lambda t_b}}(1+λt_b))}{{\lambda e^{-\lambda t_a}}}$$

$t_a$ is the time up until a device has lived

$t_b$ is point up to which I want to determine the expected lifetime.

The above formula was derived from the following where the exponential distribution is used: $$\int_{t_a}^{t_b} \frac{x f(x)}{1-F(t_a)}dx$$ $$\int_{t_a}^{t_b} \frac{x\lambda e^{-\lambda x}}{1-(1-e^{-\lambda t_a})}dx$$

My question is when $t_a$ and $t_b$ are very close to one another my expected remaining lifetime value does not lie between $t_a$ and $t_b$. This does not make any sense to me. Is there a mistake in my derivation or is my reasoning wrong that the expected remaining lifetime value should lie between $t_a$ and $t_b$. The problem is accentuated when I try and calculate the residual life when attempting to subtract $t_a$ from the mean expected lifetime value and then I obtain negative values.

For example, using the mean expected lifetime formula:

$$\int_{59}^{60} \frac{x\lambda e^{-\lambda x}}{1-(1-e^{-59\lambda})}dx$$

I would expect this answer to lie between 59 and 60. As this is the expected lifetime value given that it lived by 59 years, say. But for some reason if $\lambda$ is 2 for example is is below 59. But we already know that it has lived passed 59. So shouldn't the expected value be greater than 59.

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Your denominator is wrong.

I assume the very fact that you added a denominator to your integral means you understand that the region $[t_a, t_b]$ does not encompass the entire domain of the probability distribution of death. To make sure you're taking the expectation over a normalized distribution, you must condition it over the death being in the range in question.

However, by taking $1 - F(t_a)$ as your denominator, you're merely conditioning your distribution over death after $t_a$, but not that it dies before $t_b.$ So you're not taking a proper expectation value because you're still only integrating over region $[t_a, t_b]$ with a distribution whose domain is $[t_a, \infty).$

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  • $\begingroup$ I see. I should condition over the interval $[t_a,t_b]$ . Thank you. $\endgroup$ – Divan Cronje Sep 25 '17 at 19:28

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