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I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid.

We want to model the lifetime $T$ of a website using the density function $Pr[X > t] \approx e^{-\rho t}$. Then, according to the math I am reading,

$\begin{eqnarray} P[X > k] & = & \int_0^t Pr[X > k\,|\,T = u] \frac{d Pr[T \leq u]}{du} du \\ & \approx & \rho \int_0^t e^{-\rho u} Pr[X > k\,|\,T = u] du \\ & \approx & \rho \int_0^t e^{-\rho u} 1_{\{l(u) > k\}} du \end{eqnarray}$

if $l(u) \sim e^{\beta u}$ for some $\beta < \rho$ (Here, $\sim$ means that $l(u)$ grows as fast as $e^{\beta u}$ - some would write $l(u) = \Theta(e^{\beta u})$). $1_{\{l(u) > k\}}$ is the indicator function that is equal to $1$ if $l(u) > k$, otherwise it is $0$.

I want to understand the intermediate steps required to jump from the PDF to the indicator function.

As a bit of context, this expression comes from mathematics that tries to model the lifetime $t$ of a website. This is used to explain why the world-wide web network is (roughly) scale-free.

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    $C$ould you provide a link to the paper or whatever you are reading?2011-03-06

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