There is a simple integral equation: $$\int_0^\infty p(s) ds = 1- \frac{1}{\lambda}p(0).$$ Do you know how to solve it for $p(0)$ given we know $p(s)$ for $s>0$? Or maybe some references to methods how to solve it?
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Perhps I'm missing something but this seems to be a rather weird question: if we know $;p(s);$ then we can, in theory at least, understand that equality...otherwise we can't possibly know why it is true. – DonAntonio Apr 04 '14 at 13:56
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what are you solving for? $\lambda$ ? – Frames Catherine White Apr 04 '14 at 13:56
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for $p(0)$. Changed the question. Thanks! – cherep Apr 04 '14 at 13:59
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@cherep in terms of $\lambda$? – Guy Apr 04 '14 at 14:00
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@DonAntonio after the correction, the question makes sense now(if it possible to find a value of $p(0)$ that is, i am not really sure) – Guy Apr 04 '14 at 14:00
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Well, it still is a weird question imo since then it all is a matter of high school algebra...?? – DonAntonio Apr 04 '14 at 14:03
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@Sabyasachi in terms of $\lambda$ and all $p(s)$ for $s>0$. – cherep Apr 04 '14 at 14:04
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@DonAntonio I mean a fixed value, not the integral. Funny answer though. – Guy Apr 04 '14 at 14:04
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Or perhaps it is meant to be a differential equation...? Hard to tell – DonAntonio Apr 04 '14 at 14:04
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If you don't know the function, @Sabyasachi, how can you possibly know the integral?? – DonAntonio Apr 04 '14 at 14:04
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@DonAntonio I have no idea. Just wondering. I guess $p$ has some restrictions, which are not mentioned here. – Guy Apr 04 '14 at 14:05
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Exactly @Sabyasachi: not mentioned here . Well, in fact it seems to be worse: it is said we know the function . – DonAntonio Apr 04 '14 at 14:08
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@cherep you might want to answer that^ – Guy Apr 04 '14 at 14:09
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$$\int_0^\infty p(s) ds = 1- \frac{1}{\lambda}p(0)\implies p(0)=\lambda\left(1-\int\limits_0^\infty p(s)ds\right)\ldots\ldots?$$
DonAntonio
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@cherep, what do you mean the constant $;p(0);$ is "under the integral"?? – DonAntonio Apr 04 '14 at 14:07
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@cherep maybe you mean $p(s)$ is still under the integral? (I wonder why s? Weird choice of variable without context) – Guy Apr 04 '14 at 14:08
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@DonAntonio I mean that $p(0)$ is unknown and need to be found... The integral is from $0$ to $\infty$, thus, if we do not know $p(0)$, we cannot take the integral, can we? – cherep Apr 04 '14 at 14:10
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@cherep yes we can. Define the integral as, $$\lim_ {x\to0^+}\int_x^\infty p(s),ds$$. – Guy Apr 04 '14 at 14:12
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