Inspired by the probability calculation here, I am trying to solve a little general one:
$$\mathbb{P} (\sum_{i=1}^{m-1} A_i + \sum_{i=1}^{m} S_i < L < \sum_{i=1}^{m} A_i + \sum_{i=1}^{m+1} S_i)$$
where,
$$A_i \sim \textrm{exp}(\lambda), \; S_i \sim \textrm{exp}(\mu), \; L \sim \textrm{exp}(\lambda), \textrm{ and } \lambda \neq \mu \textrm{ are two integers}.$$
All $A_i, S_i, \textrm{ and } L$ are mutually independent. $m$ is an integer parameter.
In this calculation, I fix $m = 4$. Because $A_i \sim \textrm{exp} (\lambda)$, we have $\sum_{i=1}^{i=3} A_i \sim \textrm{Erlang}(3, \lambda)$.
Probability[
a3 + s4 < middle < a4 + s5,
{a3 \[Distributed] ErlangDistribution[3, λ],
s4 \[Distributed] ErlangDistribution[4, μ],
a4 \[Distributed] ErlangDistribution[4, λ],
s5 \[Distributed] ErlangDistribution[5, μ],
middle \[Distributed] ExponentialDistribution[λ]}]
After a few minutes, I got a long result from which I could not read anything useful:
$\frac{\mu ^4 \left(186624 \lambda ^{16}+3825792 \lambda ^{15} \mu +35971776 \lambda ^{14} \mu ^2+205589664 \lambda ^{13} \mu ^3+798109200 \lambda ^{12} \mu ^4+2217713192 \lambda ^{11} \mu ^5+4537481548 \lambda ^{10} \mu ^6+6954729890 \lambda ^9 \mu ^7+8071898695 \lambda ^8 \mu ^8+7133161040 \lambda ^7 \mu ^9+4799247376 \lambda ^6 \mu ^{10}+2441453824 \lambda ^5 \mu ^{11}+923808416 \lambda ^4 \mu ^{12}+252053248 \lambda ^3 \mu ^{13}+46855424 \lambda ^2 \mu ^{14}+5307904 \lambda \mu ^{15}+276224 \mu ^{16}\right)}{11664 (\lambda +\mu )^5 (2 \lambda +\mu )^7 (\lambda +2 \mu )^8}$
In addition, the command FullSimplify[Probability[]]
does not help.
Edit (2014-11-17):
By paper-and-pencil, I have obtained a closed form $\frac{\mu + 2 \lambda}{\mu + \lambda} (\frac{1}{2} \frac{\mu}{\mu + \lambda})^{m}$. Notice that the sum $\sum_{m=1}^{\infty} \frac{\mu + 2 \lambda}{\mu + \lambda} (\frac{1}{2} \frac{\mu}{\mu + \lambda})^{m} = \frac{\mu}{\mu + \lambda}.$ The remaining $\frac{\lambda}{\mu + \lambda}$ for special case $m = 0$ has been omitted here.
Therefore, the question is
How to get a more compact form of the above probability? Specifically, is it consistent with my manual calculation (which may be wrong)?
a3
anda4
,s4
ands5
are independent, but actually they're not,a3
is part ofa4
,s4
is part ofs5
! $\endgroup$