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Let

  • $a, b$ be two variables drawn from an exponential distribution with parameter $\lambda_1$.

  • $c, d$ be two variables drawn from an exponential distribution with parameter $\lambda_2$.

I am interested in the probability density function (PDF) of random variable

$$\frac{ac+bd}{c+d}$$

Simulations give the same PDF of that of the random variable $\frac{a+b}{2}$, which is the exponential distribution with parameter $\lambda_1$.

I've spent a month trying to prove it but without success. Could you please help by proving whether this statement holds true or not?

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    $\begingroup$ OK. apparently, they are very similar but not identical. $\endgroup$ – Chrysanthi Pas Jul 16 '19 at 12:57
  • $\begingroup$ Ok, apparently, they are not the same :( $\endgroup$ – Chrysanthi Pas Jul 16 '19 at 12:57
  • $\begingroup$ Do you assume independence? $\endgroup$ – kjetil b halvorsen Jul 16 '19 at 13:03
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    $\begingroup$ Just posting a question on MO seems to be very enlightening! $\endgroup$ – Jochen Wengenroth Jul 16 '19 at 13:12
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    $\begingroup$ It’s a good lesson to learn, and something that happens to everyone at least once. When you are spending a lot of time trying to prove something, always remember it may not be true! $\endgroup$ – Dan Romik Jul 16 '19 at 18:29
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I'm assuming you mean $a, b, c, d$ to be independent exponential random variables with rate parameters $\lambda_1, \lambda_1, \lambda_2, \lambda_2$.

I find that $(ac+bd)/(c+d)$ has mean $\lambda_1^{-1}$ (the same as $(a+b)/2$), but variance $2 \lambda_1^{-2}/3$ while $(a+b)/2$ has variance $\lambda_1^{-2}/2$.

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$\newcommand{\la}{\lambda}$ Welcome to MathOverflow!

This conjecture is of course false, if the informal term "drawn" you are using means independence. Indeed, by homogeneity, without loss of generality $\la_2=1$. Let now $\la:=\la_1$, $(U,V,X,Y):=(a,b,c,d)$, \begin{equation*} S:=\frac{UX+VY}{X+Y},\quad T:=\frac{U+V}2. \end{equation*}

We have to show that the pdfs $f_S$ and $f_T$ of $S$ and $T$ differ from each other. The random variable (r.v.) $T$ has the Gamma distribution with parameters $2,\la/2$. So, \begin{equation*} f_T(t)=4t\,e^{-2t/\la}/\la^21_{t>0}, \end{equation*} whence \begin{equation*} f_T(t)/t\to4/\la^2\quad\text{ as }\quad t\downarrow0. \end{equation*}

On the other hand, solving the equation $s=\frac{ux+vy}{x+y}$ for $u$, to get $u=\frac{(x+y)s-vy}x$ and $\frac{\partial u}{\partial s}=\frac{x+y}x$, we see that the joint pdf of $(S,V,X,Y)$ is given by \begin{align*} f_{S,V,X,Y}(s,v,x,y)&=f_{U,V,X,Y}(\tfrac{(x+y)s-vy}x,v,x,y)\frac{x+y}x \\ &=f_U(\tfrac{(x+y)s-vy}x)f_V(v)f_X(x)f_Y(y)\frac{x+y}x \\ &=\frac1{\la^2}\,\frac{x+y}x\, \exp\Big\{\frac{vy-(x+y)s}{\la x}-\frac v\la-x-y\Big\} 1_{x,y,v>0,\ s>\frac{vy}{x+y}}. \end{align*} Next, for the pdf of $S$ we have \begin{align*} f_S(s)&=\iint_{x,y>0} dx\,dy\,\int_0^\infty dv\,f_{S,V,X,Y}(s,v,x,y) \\ &=\frac1{\la^2}\,\iint_{x,y>0} dx\,dy\,\frac{x+y}x\,\exp\Big\{\frac{-(x+y)s}{\la x}-x-y\Big\}\int_0^{(x+y)s/y} dv\,\exp\Big\{\frac{vy}{\la x}-\frac v\la \Big\} \\ &=\frac s{\la^2}\,\iint_{x,y>0} dx\,dy\,e^{-x-y} \frac{(x+y)^2 }{x y}\,r\left(\frac{x+y}{\la y},\frac{x+y}{\la x},s\right), \end{align*} where \begin{equation*} r(a,b,s)=\frac{e^{-a s}-e^{-b s}}{(b-a)s} =\frac1{b-a}\,\int_a^b dz\,e^{-z s} \end{equation*} for positive real $b\ne a$, so that $r(a,b,s)$ is decreasing in $s$; as usual, here we let $\int_a^b:=-\int_b^a$ if $b<a$; also, $r(a,b,s)\uparrow1$ as $s\downarrow0$. So, by the monotone convergence theorem, \begin{align*} f_S(s)/s&\to\frac1{\la^2}\,\iint_{x,y>0} dx\,dy\,e^{-x-y} \frac{(x+y)^2 }{x y} \\ &\ge\frac1{\la^2}\,\iint_{x,y>0} dx\,dy\,e^{-x-y} \frac xy=\infty \end{align*} as $s\downarrow0$. Comparing this with (1), we see $f_S\ne f_T$, as claimed.

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The command of Mathematica 12.2

PDF[TransformedDistribution[(a*c + b*d)/(c + d), {a \[Distributed] 
ExponentialDistribution[\[Lambda]1], b \[Distributed] ExponentialDistribution[\[Lambda]1], 
c \[Distributed] ExponentialDistribution[\[Lambda]2],d \[Distributed] ExponentialDistribution[\[Lambda]2]}], t]

answers $$\begin{cases} \frac{1}{2} \text{$\lambda $1} \left(-\text{Ei}(-t \text{$\lambda $1})+2 e^{-2 \text{$\lambda $1} t} \text{Ei}(t \text{$\lambda $1})+\Gamma (0,t \text{$\lambda $1})\right) & t>0 \\ 0 & t<0 \\ \text{Indeterminate} & \text{True} \end{cases} $$

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  • $\begingroup$ It should be noticed that the PDF under consideration does not depend on $\lambda2$. $\endgroup$ – user64494 Jan 8 at 20:48

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