Explaining the Joint Distribution of T1,T2,...,Tn given N(t)=n

[kingwinner]

Let {N(t): t≥0} be a Poisson process of rate λ.
We are given that for a fixed t, N(t)=n.
Let T_i be the time of the i^th event, i=1,2,...,n.

Then the event {T₁≤t₁, T₂≤t₂,...,T_n≤t_n, and N(t)=n} occurs if and only if exactly one event occurs in each of the intervals [0,t₁], (t₁,t₂],..., (t_n-1,t_n], and no events occur in (t_n,t].
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I don't understand the 'exactly one' part.
For example, T₂≤t₂ just says that T₂ is less than or equal to t₂, and T₂ can very possibly be less than t₁ as well, right? (since it did NOT say that T₂ MUST be larger than t₁) In this case, we would then have more than one event occurring in [0,t₁]. Why is this not allowed? I don't get it...

Can someone please explain? I would really appreciate it!

 

[EnumaElish]

Would N(t)=n be satisfied if T2 < t1?

 

[EnumaElish]

It seems to be a definitional choice that serves some purpose in the rest of the problem.

 

[kingwinner]

Would N(t)=n be satisfied if T2 < t1?

I think so! We can possibly have all n points in the interval [0,t1].

 

[kingwinner]

The book is trying to prove that the joint density function of T1,T2,...,Tn given that N(t)=n is given by (n!)/(t^n), 0<t1<t2<...<t_n<t

They are first trying to find the joint distribution function for 0<t1<t2<...<t_n<t.
i.e. P(T1≤t1,...Tn≤tn |N(t)=n)
and they commented that "the event {T₁≤t₁, T₂≤t₂,...,T_n≤t_n, and N(t)=n} occurs if and only if exactly one event occurs in each of the intervals [0,t₁], (t₁,t₂],..., (t_n-1,t_n], and no events occur in (t_n,t". This is where I got totally confused...

Can someone please help?