- #1
Mehmood_Yasir
- 68
- 2
Assume a Poisson process with rate ##\lambda##.
Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time ##1^{st}, 2^{nd}, 3^{rd}... ## i.e.,
1. ##E[T_{1}|T_{1}\le T]## ?
2. ##E[T_{2}|T_{1}<T_{2}\le T]## ?
3. ##E[T_{3}|T_{2}<T_{3}\le T]## ?
My approach! For Poisson process with rate ##\lambda##, each time interval corresponds to a random variable ##X_i## with an exponeitial distribution.Therefore,
\begin{align*}
&T_1=X_1 \\&
T_2=X_1+X_2\\&T_3=X_1+X_2+X3\\&...
\end{align*}
##T_i## has Gamma distribuiotn ##\Gamma(i,\lambda)##. If ##T## is a deterministic value not a random variable. Then, ##E[T_{1}|T_{1}\le T]##
\begin{align*}
E[T_{1}|T_{1}\le T]&=\int_{0}^\infty t_1f_{T_1|T_1 \le T}(t_1)dt_1\\&=\frac{1}{1-e^{-\lambda T}}\int_{0}^T t_1 \lambda e^{-\lambda t_1}dt_1\\&=\frac{-1}{1-e^{-\lambda T}}\int_0^T t_1de^{-\lambda t_1}\\&\text{(Final solution is )}\\&=\frac{-(Te^{-\lambda T}+\frac{1}{\lambda}e^{-\lambda T}-\frac{1}{\lambda})}{1-e^{-\lambda T}}
\end{align*}
What about ##E[T_{2}|T_{1}<T_{2}\le T]## and ##E[T_{3}|T_{2}<T_{3}\le T]##.
Can someone please guide me? I thank in advance.
Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time ##1^{st}, 2^{nd}, 3^{rd}... ## i.e.,
1. ##E[T_{1}|T_{1}\le T]## ?
2. ##E[T_{2}|T_{1}<T_{2}\le T]## ?
3. ##E[T_{3}|T_{2}<T_{3}\le T]## ?
My approach! For Poisson process with rate ##\lambda##, each time interval corresponds to a random variable ##X_i## with an exponeitial distribution.Therefore,
\begin{align*}
&T_1=X_1 \\&
T_2=X_1+X_2\\&T_3=X_1+X_2+X3\\&...
\end{align*}
##T_i## has Gamma distribuiotn ##\Gamma(i,\lambda)##. If ##T## is a deterministic value not a random variable. Then, ##E[T_{1}|T_{1}\le T]##
\begin{align*}
E[T_{1}|T_{1}\le T]&=\int_{0}^\infty t_1f_{T_1|T_1 \le T}(t_1)dt_1\\&=\frac{1}{1-e^{-\lambda T}}\int_{0}^T t_1 \lambda e^{-\lambda t_1}dt_1\\&=\frac{-1}{1-e^{-\lambda T}}\int_0^T t_1de^{-\lambda t_1}\\&\text{(Final solution is )}\\&=\frac{-(Te^{-\lambda T}+\frac{1}{\lambda}e^{-\lambda T}-\frac{1}{\lambda})}{1-e^{-\lambda T}}
\end{align*}
What about ##E[T_{2}|T_{1}<T_{2}\le T]## and ##E[T_{3}|T_{2}<T_{3}\le T]##.
Can someone please guide me? I thank in advance.