Undergrad Conditional Expectation Value of Poisson Arrival in Fixed T

[Mehmood_Yasir]

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.

 

[haruspex]

I think this is valid:
Suppose there are exactly m events in [0,T]. The probability of that is P_T,m=$\frac{(\lambda T)^m}{m!}e^{-\lambda T}$.
For n≤m, the expected time to the n^th of them is $E[n,m,T]=\frac{nT}{m+1}$. The expected time over all values of m is then $\frac{\Sigma _{m≥n}E[n,m,T]P_{T,m}}{\Sigma _{m≥n} P_{T,m}}$.

As a check, for large n that approximates $\frac{nT}{n+1}$, as it should.

 

[Mehmood_Yasir]

@haruspex many thanks for your answer. I tried to understand it but could not get it completely. Actually we don't know the no. of events in time $[0-T]$.
Let me put this way. E.g., assume an arrival appear at $t=0$ and starts a clock for a duration of Time $T$. Now the rest of the arrivals follow poisson process with rate $\lambda$. When clock time which is $T$ expires, we have to calculate now the expected arrival time of the first, second and third,...so on.
I did for the first arrival which is in fact conditional expectation value of the arrival time i.e., $E[T_{1}|T_{1}<=T]$.

Since the expected value of the arrival time of the second arrival will be greater than the arrival time of the first arrival and will be less than $T$. Is this approach CORRECT analogous to my first proposal for determining value of the $E[T_{2}|T_{1}<T_{2}<=T]$ (see the attached image)?

 

[haruspex]

Actually we don't know the no. of events in time [0−T]

I understand that. If you look at my final expression you will see it sums over all m (total number of events in T) ≥ n.