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I have a probability distribution $f(x) = c\cdot\exp(-c\cdot x)$ where $c\in\Bbb R$ and $ x \in [0, \infty ) $

I have to make a collection of 5 randomly selected values from this distribution. $ x_1,x_2,x_3,x_4, x_5 $ which are randomly selected based on the probability distribution. (I would expect more frequent values at small $x$ because its more probable.. exponential decay of probability).

Now let's say $ n = x_1 + x_2 + x_3+ x_4 + x_5 $

What would be the new probability distribution for the collection wrt 'n' i.e. p(n)?

I am not sure how to start. I understand that I could use multiplication of probabilities for a 'n': $ f(x_1)\cdot f(x_2)\cdots f(x_5) $ but how to include condition of $ x_1+x_2+x_3+x_4+x_5=n $ to find p$(n)$?

Graham Kemp
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Vipul Tomar
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  • Consider typing up the math in MathJax, makes it much more readable. – Richard Jensen Sep 09 '22 at 11:15
  • What you have there is an exponential distribution with density $f(x)$ and intensity $c$. What you seek is the distribution of the $n$-th jump time of a Poisson process. See this post how to deal with such problems. – Kurt G. Sep 09 '22 at 11:20
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    Note that $f$ is not a probability density on $[0,N]$ as the integral of $f$ over this interval is strictly smaller than $1$. If you consider $f$ on the interval $[0,\infty)$ instead you have the exponential distribution with rate $c$. It is a special case of the Gamma distribution. – jakobdt Sep 09 '22 at 11:21
  • @jakobdt I am basically doing it for [0,infinity), with rate c and I thought it should have area under the curve as 1, in that case. Right? – Vipul Tomar Sep 09 '22 at 11:33
  • @RichardJensen Sorry. I am still looking up, how to format it with MathJax. – Vipul Tomar Sep 09 '22 at 11:34
  • @VipulTomar Here's an old guide, but the basics should be the same. – Richard Jensen Sep 09 '22 at 12:15
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    @RichardJensen Thank you.. Working on it – Vipul Tomar Sep 09 '22 at 12:28
  • I see that it is correct that $f(x)$ is not a probability distribution on $[0,N)$. @VipulTomar What is the point of having this $N$ at all ? – Kurt G. Sep 09 '22 at 12:47
  • Okay. I edited the question. I replaced N to infinity. Hopefully it makes more sense. – Vipul Tomar Sep 09 '22 at 12:53
  • The class of exponential distributions is contained in the class of Gamma distributions. What is particularly nice about the latter is that the sum $X_1+\dotsm+X_n$ of $n$ independent Gamma distributed random variables $X_1,\dotsc,X_n$ is again Gamma distributed. – jakobdt Sep 09 '22 at 13:06
  • Thank you all for the suggestions. Looks like I was making some huge mistake. So I am looking at it in a different way. So I will post as a new question to not create confusion. – Vipul Tomar Sep 09 '22 at 15:59

1 Answers1

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We have independence of the samples, and that they are identically exponentially distributed with rate parameter $c$.

Apply the Law of Total Probability.

When five non-negative values sum to a non-negative total ($n$), then four have to sum to at most $n$, and the fifth shall be the remainder.

$$\begin{align} p(n) &= \mathbf 1_{0\leqslant n}\cdot{\mathop{\iint\!\!\!\iint}\limits_{s+t+u+v\leq n} f(s)f(t)f(u)f(v)f(n-s-t-u-v)\,\mathrm d v\,\mathrm d u\,\mathrm d t\,\mathrm d s}\\&=\mathbf 1_{0\leq n}\cdot\int_0^n\int_0^{n-s}\int_0^{n-s-t}\int_0^{n-s-t-u}c^5\mathrm e^{-c n}\,\mathrm d v\,\mathrm d u\,\mathrm d t\,\mathrm d s\\&~~\vdots\end{align}$$

Graham Kemp
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