Distribution of sum of independent exponentials with random number of summands












3












$begingroup$


Let $tau_isimexpleft(lambdaright)$ be independent and identically distributed exponentials with parameter $lambda$. Then, for given $n$, the sum of these values
$$T_n := sum_{i=0}^n tau_i$$
follows an Erlang-Distribution with probability density function
$$pi(T_n=T| n,lambda)={lambda^n T^{n-1} e^{-lambda T} over (n-1)!}quadmbox{for }T, lambda geq 0.$$



I am interested in the distribution of $T_tilde n$ where $tilde n$ is a random variable such that for $tau_a sim exp(lambda_a)$ exponentially distributed, it holds that
$$T_tilde n leq tau_a \T_{tilde{n}+1} > tau_a.$$



In other words, $T_{tilde n}$ is truncated by an exponential distribution. I fail in deriving the distribution of $tilde n$ but maybe there is an easier way:
$$pileft(tilde n = kright) = pileft(T_n < tau_a|n=kright)\
=1-intlimits_{mathbb{R}^+}sumlimits_{n=0}^{k-1}frac{1}{n!}expleft(-(lambda+lambda_a)tau_aright)(taulambda_a)^nlambda_adtau_a.$$



However, just sampling and eye-balling looks to me like this density isn't that ugly:



iter <- 20000

lambda_a <- 1
lambda <- 2

df <- data.frame(tau=rep(NA, iter), a=rep(NA, iter))

for(i in 1:iter){
set.seed(i)
a <- rexp(1, rate = lambda_a)
s <- cumsum(rexp(500, rate = lambda))

df[i,] <- c(max(s[1], s[s<a]), a)
}

library(tidyverse)

ggplot(df %>% gather(), aes(x = value, fill = key)) +
geom_density(alpha = .3) + theme_bw()


enter image description here










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$endgroup$








  • 1




    $begingroup$
    I would recommend not using the same notation for both $tau_i$ and the sum $tau_n$.
    $endgroup$
    – ukemi
    3 hours ago


















3












$begingroup$


Let $tau_isimexpleft(lambdaright)$ be independent and identically distributed exponentials with parameter $lambda$. Then, for given $n$, the sum of these values
$$T_n := sum_{i=0}^n tau_i$$
follows an Erlang-Distribution with probability density function
$$pi(T_n=T| n,lambda)={lambda^n T^{n-1} e^{-lambda T} over (n-1)!}quadmbox{for }T, lambda geq 0.$$



I am interested in the distribution of $T_tilde n$ where $tilde n$ is a random variable such that for $tau_a sim exp(lambda_a)$ exponentially distributed, it holds that
$$T_tilde n leq tau_a \T_{tilde{n}+1} > tau_a.$$



In other words, $T_{tilde n}$ is truncated by an exponential distribution. I fail in deriving the distribution of $tilde n$ but maybe there is an easier way:
$$pileft(tilde n = kright) = pileft(T_n < tau_a|n=kright)\
=1-intlimits_{mathbb{R}^+}sumlimits_{n=0}^{k-1}frac{1}{n!}expleft(-(lambda+lambda_a)tau_aright)(taulambda_a)^nlambda_adtau_a.$$



However, just sampling and eye-balling looks to me like this density isn't that ugly:



iter <- 20000

lambda_a <- 1
lambda <- 2

df <- data.frame(tau=rep(NA, iter), a=rep(NA, iter))

for(i in 1:iter){
set.seed(i)
a <- rexp(1, rate = lambda_a)
s <- cumsum(rexp(500, rate = lambda))

df[i,] <- c(max(s[1], s[s<a]), a)
}

library(tidyverse)

ggplot(df %>% gather(), aes(x = value, fill = key)) +
geom_density(alpha = .3) + theme_bw()


enter image description here










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I would recommend not using the same notation for both $tau_i$ and the sum $tau_n$.
    $endgroup$
    – ukemi
    3 hours ago
















3












3








3


1



$begingroup$


Let $tau_isimexpleft(lambdaright)$ be independent and identically distributed exponentials with parameter $lambda$. Then, for given $n$, the sum of these values
$$T_n := sum_{i=0}^n tau_i$$
follows an Erlang-Distribution with probability density function
$$pi(T_n=T| n,lambda)={lambda^n T^{n-1} e^{-lambda T} over (n-1)!}quadmbox{for }T, lambda geq 0.$$



I am interested in the distribution of $T_tilde n$ where $tilde n$ is a random variable such that for $tau_a sim exp(lambda_a)$ exponentially distributed, it holds that
$$T_tilde n leq tau_a \T_{tilde{n}+1} > tau_a.$$



In other words, $T_{tilde n}$ is truncated by an exponential distribution. I fail in deriving the distribution of $tilde n$ but maybe there is an easier way:
$$pileft(tilde n = kright) = pileft(T_n < tau_a|n=kright)\
=1-intlimits_{mathbb{R}^+}sumlimits_{n=0}^{k-1}frac{1}{n!}expleft(-(lambda+lambda_a)tau_aright)(taulambda_a)^nlambda_adtau_a.$$



However, just sampling and eye-balling looks to me like this density isn't that ugly:



iter <- 20000

lambda_a <- 1
lambda <- 2

df <- data.frame(tau=rep(NA, iter), a=rep(NA, iter))

for(i in 1:iter){
set.seed(i)
a <- rexp(1, rate = lambda_a)
s <- cumsum(rexp(500, rate = lambda))

df[i,] <- c(max(s[1], s[s<a]), a)
}

library(tidyverse)

ggplot(df %>% gather(), aes(x = value, fill = key)) +
geom_density(alpha = .3) + theme_bw()


enter image description here










share|cite|improve this question











$endgroup$




Let $tau_isimexpleft(lambdaright)$ be independent and identically distributed exponentials with parameter $lambda$. Then, for given $n$, the sum of these values
$$T_n := sum_{i=0}^n tau_i$$
follows an Erlang-Distribution with probability density function
$$pi(T_n=T| n,lambda)={lambda^n T^{n-1} e^{-lambda T} over (n-1)!}quadmbox{for }T, lambda geq 0.$$



I am interested in the distribution of $T_tilde n$ where $tilde n$ is a random variable such that for $tau_a sim exp(lambda_a)$ exponentially distributed, it holds that
$$T_tilde n leq tau_a \T_{tilde{n}+1} > tau_a.$$



In other words, $T_{tilde n}$ is truncated by an exponential distribution. I fail in deriving the distribution of $tilde n$ but maybe there is an easier way:
$$pileft(tilde n = kright) = pileft(T_n < tau_a|n=kright)\
=1-intlimits_{mathbb{R}^+}sumlimits_{n=0}^{k-1}frac{1}{n!}expleft(-(lambda+lambda_a)tau_aright)(taulambda_a)^nlambda_adtau_a.$$



However, just sampling and eye-balling looks to me like this density isn't that ugly:



iter <- 20000

lambda_a <- 1
lambda <- 2

df <- data.frame(tau=rep(NA, iter), a=rep(NA, iter))

for(i in 1:iter){
set.seed(i)
a <- rexp(1, rate = lambda_a)
s <- cumsum(rexp(500, rate = lambda))

df[i,] <- c(max(s[1], s[s<a]), a)
}

library(tidyverse)

ggplot(df %>% gather(), aes(x = value, fill = key)) +
geom_density(alpha = .3) + theme_bw()


enter image description here







distributions exponential-family truncation






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edited 3 hours ago







muffin1974

















asked 4 hours ago









muffin1974muffin1974

519419




519419








  • 1




    $begingroup$
    I would recommend not using the same notation for both $tau_i$ and the sum $tau_n$.
    $endgroup$
    – ukemi
    3 hours ago
















  • 1




    $begingroup$
    I would recommend not using the same notation for both $tau_i$ and the sum $tau_n$.
    $endgroup$
    – ukemi
    3 hours ago










1




1




$begingroup$
I would recommend not using the same notation for both $tau_i$ and the sum $tau_n$.
$endgroup$
– ukemi
3 hours ago






$begingroup$
I would recommend not using the same notation for both $tau_i$ and the sum $tau_n$.
$endgroup$
– ukemi
3 hours ago












1 Answer
1






active

oldest

votes


















3












$begingroup$

As detailed in this X validated answer, waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed one produces an Poisson $mathcal P(lambda)$ variate $N$. Hence waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed $tau_a$ produces an Poisson $mathcal P(tau_alambda)$ variate $N$, conditional on $tau_a$. Therefore
begin{align*}
mathbb P(N=n)&=int_0^infty mathbb P(N=n|tau_a) ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&= int_0^infty dfrac{(lambdatau_a)^n}{n!},e^{-tau_alambda} ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},int_0^infty tau_a^n,e^{-tau_a(lambda+lambda_a)} ,text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},dfrac{Gamma(n+1)}{(lambda_a+lambda)^{n+1}}=dfrac{lambda_alambda^n}{(lambda_a+lambda)^{n+1}}
end{align*}

which is a Geometric $mathcal G(lambda_a/{lambda_a+lambda})$ random variable.



Considering now the distribution of$$zeta=sum_{i=1}^N tau_i$$the moment generating function of $zeta$ is
$$mathbb E[e^{zzeta}]=mathbb E[e^{z{tau_1+cdots+tau_N}}]=mathbb E^N[mathbb E^{tau_1}[e^{ztau_1}]^N]=E^N[{lambda/(lambda-z)}^N]=E^N[e^{N(ln lambda-ln (lambda-z))}]$$and the mgf of a Geometric $mathcal G(p)$ variate is
$$varphi_N(z)=dfrac{pe^z}{1-(1-p)e^z}$$Hence the moment generating function of $zeta$ is $$dfrac{pe^{ln lambda-ln (lambda-z)}}{1-(1-(lambda_a/{lambda_a+lambda}))e^{ln lambda-ln (lambda-z)}}=dfrac{p lambda}{ lambda-z-lambda^2/{lambda_a+lambda}}$$
modulo typing mistakes






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks a lot @Xi'an! Do I get it right that in your notation $p= lambda_a/(lambda_a+lambda)$ ? Because then the moment generating function in the last line is equivalent to $frac{1}{1-zleft(plambdaright)^{-1}}$ which corresponds to the MGF of an exponential distribution..
    $endgroup$
    – muffin1974
    2 hours ago











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









3












$begingroup$

As detailed in this X validated answer, waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed one produces an Poisson $mathcal P(lambda)$ variate $N$. Hence waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed $tau_a$ produces an Poisson $mathcal P(tau_alambda)$ variate $N$, conditional on $tau_a$. Therefore
begin{align*}
mathbb P(N=n)&=int_0^infty mathbb P(N=n|tau_a) ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&= int_0^infty dfrac{(lambdatau_a)^n}{n!},e^{-tau_alambda} ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},int_0^infty tau_a^n,e^{-tau_a(lambda+lambda_a)} ,text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},dfrac{Gamma(n+1)}{(lambda_a+lambda)^{n+1}}=dfrac{lambda_alambda^n}{(lambda_a+lambda)^{n+1}}
end{align*}

which is a Geometric $mathcal G(lambda_a/{lambda_a+lambda})$ random variable.



Considering now the distribution of$$zeta=sum_{i=1}^N tau_i$$the moment generating function of $zeta$ is
$$mathbb E[e^{zzeta}]=mathbb E[e^{z{tau_1+cdots+tau_N}}]=mathbb E^N[mathbb E^{tau_1}[e^{ztau_1}]^N]=E^N[{lambda/(lambda-z)}^N]=E^N[e^{N(ln lambda-ln (lambda-z))}]$$and the mgf of a Geometric $mathcal G(p)$ variate is
$$varphi_N(z)=dfrac{pe^z}{1-(1-p)e^z}$$Hence the moment generating function of $zeta$ is $$dfrac{pe^{ln lambda-ln (lambda-z)}}{1-(1-(lambda_a/{lambda_a+lambda}))e^{ln lambda-ln (lambda-z)}}=dfrac{p lambda}{ lambda-z-lambda^2/{lambda_a+lambda}}$$
modulo typing mistakes






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks a lot @Xi'an! Do I get it right that in your notation $p= lambda_a/(lambda_a+lambda)$ ? Because then the moment generating function in the last line is equivalent to $frac{1}{1-zleft(plambdaright)^{-1}}$ which corresponds to the MGF of an exponential distribution..
    $endgroup$
    – muffin1974
    2 hours ago
















3












$begingroup$

As detailed in this X validated answer, waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed one produces an Poisson $mathcal P(lambda)$ variate $N$. Hence waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed $tau_a$ produces an Poisson $mathcal P(tau_alambda)$ variate $N$, conditional on $tau_a$. Therefore
begin{align*}
mathbb P(N=n)&=int_0^infty mathbb P(N=n|tau_a) ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&= int_0^infty dfrac{(lambdatau_a)^n}{n!},e^{-tau_alambda} ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},int_0^infty tau_a^n,e^{-tau_a(lambda+lambda_a)} ,text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},dfrac{Gamma(n+1)}{(lambda_a+lambda)^{n+1}}=dfrac{lambda_alambda^n}{(lambda_a+lambda)^{n+1}}
end{align*}

which is a Geometric $mathcal G(lambda_a/{lambda_a+lambda})$ random variable.



Considering now the distribution of$$zeta=sum_{i=1}^N tau_i$$the moment generating function of $zeta$ is
$$mathbb E[e^{zzeta}]=mathbb E[e^{z{tau_1+cdots+tau_N}}]=mathbb E^N[mathbb E^{tau_1}[e^{ztau_1}]^N]=E^N[{lambda/(lambda-z)}^N]=E^N[e^{N(ln lambda-ln (lambda-z))}]$$and the mgf of a Geometric $mathcal G(p)$ variate is
$$varphi_N(z)=dfrac{pe^z}{1-(1-p)e^z}$$Hence the moment generating function of $zeta$ is $$dfrac{pe^{ln lambda-ln (lambda-z)}}{1-(1-(lambda_a/{lambda_a+lambda}))e^{ln lambda-ln (lambda-z)}}=dfrac{p lambda}{ lambda-z-lambda^2/{lambda_a+lambda}}$$
modulo typing mistakes






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks a lot @Xi'an! Do I get it right that in your notation $p= lambda_a/(lambda_a+lambda)$ ? Because then the moment generating function in the last line is equivalent to $frac{1}{1-zleft(plambdaright)^{-1}}$ which corresponds to the MGF of an exponential distribution..
    $endgroup$
    – muffin1974
    2 hours ago














3












3








3





$begingroup$

As detailed in this X validated answer, waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed one produces an Poisson $mathcal P(lambda)$ variate $N$. Hence waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed $tau_a$ produces an Poisson $mathcal P(tau_alambda)$ variate $N$, conditional on $tau_a$. Therefore
begin{align*}
mathbb P(N=n)&=int_0^infty mathbb P(N=n|tau_a) ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&= int_0^infty dfrac{(lambdatau_a)^n}{n!},e^{-tau_alambda} ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},int_0^infty tau_a^n,e^{-tau_a(lambda+lambda_a)} ,text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},dfrac{Gamma(n+1)}{(lambda_a+lambda)^{n+1}}=dfrac{lambda_alambda^n}{(lambda_a+lambda)^{n+1}}
end{align*}

which is a Geometric $mathcal G(lambda_a/{lambda_a+lambda})$ random variable.



Considering now the distribution of$$zeta=sum_{i=1}^N tau_i$$the moment generating function of $zeta$ is
$$mathbb E[e^{zzeta}]=mathbb E[e^{z{tau_1+cdots+tau_N}}]=mathbb E^N[mathbb E^{tau_1}[e^{ztau_1}]^N]=E^N[{lambda/(lambda-z)}^N]=E^N[e^{N(ln lambda-ln (lambda-z))}]$$and the mgf of a Geometric $mathcal G(p)$ variate is
$$varphi_N(z)=dfrac{pe^z}{1-(1-p)e^z}$$Hence the moment generating function of $zeta$ is $$dfrac{pe^{ln lambda-ln (lambda-z)}}{1-(1-(lambda_a/{lambda_a+lambda}))e^{ln lambda-ln (lambda-z)}}=dfrac{p lambda}{ lambda-z-lambda^2/{lambda_a+lambda}}$$
modulo typing mistakes






share|cite|improve this answer









$endgroup$



As detailed in this X validated answer, waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed one produces an Poisson $mathcal P(lambda)$ variate $N$. Hence waiting for a sum of iid exponential $mathcal E(lambda)$ variates to exceed $tau_a$ produces an Poisson $mathcal P(tau_alambda)$ variate $N$, conditional on $tau_a$. Therefore
begin{align*}
mathbb P(N=n)&=int_0^infty mathbb P(N=n|tau_a) ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&= int_0^infty dfrac{(lambdatau_a)^n}{n!},e^{-tau_alambda} ,lambda_a e^{-lambda_atau_a},text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},int_0^infty tau_a^n,e^{-tau_a(lambda+lambda_a)} ,text{d}tau_a\
&=dfrac{lambda_alambda^n}{n!},dfrac{Gamma(n+1)}{(lambda_a+lambda)^{n+1}}=dfrac{lambda_alambda^n}{(lambda_a+lambda)^{n+1}}
end{align*}

which is a Geometric $mathcal G(lambda_a/{lambda_a+lambda})$ random variable.



Considering now the distribution of$$zeta=sum_{i=1}^N tau_i$$the moment generating function of $zeta$ is
$$mathbb E[e^{zzeta}]=mathbb E[e^{z{tau_1+cdots+tau_N}}]=mathbb E^N[mathbb E^{tau_1}[e^{ztau_1}]^N]=E^N[{lambda/(lambda-z)}^N]=E^N[e^{N(ln lambda-ln (lambda-z))}]$$and the mgf of a Geometric $mathcal G(p)$ variate is
$$varphi_N(z)=dfrac{pe^z}{1-(1-p)e^z}$$Hence the moment generating function of $zeta$ is $$dfrac{pe^{ln lambda-ln (lambda-z)}}{1-(1-(lambda_a/{lambda_a+lambda}))e^{ln lambda-ln (lambda-z)}}=dfrac{p lambda}{ lambda-z-lambda^2/{lambda_a+lambda}}$$
modulo typing mistakes







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 3 hours ago









Xi'anXi'an

57.2k895360




57.2k895360












  • $begingroup$
    Thanks a lot @Xi'an! Do I get it right that in your notation $p= lambda_a/(lambda_a+lambda)$ ? Because then the moment generating function in the last line is equivalent to $frac{1}{1-zleft(plambdaright)^{-1}}$ which corresponds to the MGF of an exponential distribution..
    $endgroup$
    – muffin1974
    2 hours ago


















  • $begingroup$
    Thanks a lot @Xi'an! Do I get it right that in your notation $p= lambda_a/(lambda_a+lambda)$ ? Because then the moment generating function in the last line is equivalent to $frac{1}{1-zleft(plambdaright)^{-1}}$ which corresponds to the MGF of an exponential distribution..
    $endgroup$
    – muffin1974
    2 hours ago
















$begingroup$
Thanks a lot @Xi'an! Do I get it right that in your notation $p= lambda_a/(lambda_a+lambda)$ ? Because then the moment generating function in the last line is equivalent to $frac{1}{1-zleft(plambdaright)^{-1}}$ which corresponds to the MGF of an exponential distribution..
$endgroup$
– muffin1974
2 hours ago




$begingroup$
Thanks a lot @Xi'an! Do I get it right that in your notation $p= lambda_a/(lambda_a+lambda)$ ? Because then the moment generating function in the last line is equivalent to $frac{1}{1-zleft(plambdaright)^{-1}}$ which corresponds to the MGF of an exponential distribution..
$endgroup$
– muffin1974
2 hours ago


















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