poisson processes help

#1
Hi, i'm stuck on a basic poisson process question. If Xt (where t is a subscript) is a poisson process with intensity lambda what is the conditional distribution of Xt1 = N(0,t1] for t>t1? I think N(0,t1] denotes the number of arrivals from 0 to t1. Helpwould be appreciated as I don't even know where to start.
 

BGM

TS Contributor
#2
You need to calculate the conditional probability mass function for \( X_{t_1}|X_t = n \)

First you use the definition of the conditional probability.

Second is to note that Poisson process has independent increments.

Third is to know the distributions of the increments, and substitute the corresponding probability mass function and simplify it.

At last you will obtain a familiar probability mass function, which you can recognize the distribution.
 
#3
You need to calculate the conditional probability mass function for \( X_{t_1}|X_t = n \)

First you use the definition of the conditional probability.

Second is to note that Poisson process has independent increments.

Third is to know the distributions of the increments, and substitute the corresponding probability mass function and simplify it.

At last you will obtain a familiar probability mass function, which you can recognize the distribution.
See this is the way i approached the problem, but on the numerator you get the probability of Xt1=m multiplied by Xt=n, and on the denominator you get probability of Xt=n so does that not cancel with Xt=n in the numerator to just leave the same pmf as I wouldv'e got without conditioning on Xt=n??
 

BGM

TS Contributor
#4
but on the numerator you get the probability of Xt1=m multiplied by Xt=n
I say the increment are independent only for non-overlapping intervals.

The interval \( (0, t_1] \) and \( (0, t] \) are overlapping intervals and therefore they are dependent. So your statement is wrong.
 
#5
I say the increment are independent only for non-overlapping intervals.

The interval \( (0, t_1] \) and \( (0, t] \) are overlapping intervals and therefore they are dependent. So your statement is wrong.
Okay well if I try follow you're steps, you say to know the distribution of the increments? I can't figure out what this distribution is, any tips?

EDIT: is it poisson(lambda(t-t1))?
 

BGM

TS Contributor
#6
Big hints: \( (0, t_1] \) and \( (t_1, t] \) are non-overlapping intervals.

I suppose you know the distribution of \( X_t - X_s \) ?

EDIT: is it poisson(lambda(t-t1))?
seems you get it
 
#7
Big hints: \( (0, t_1] \) and \( (t_1, t] \) are non-overlapping intervals.

I suppose you know the distribution of \( X_t - X_s \) ?



seems you get it

Thanks for your help, by the way how do you type using actual maths notation?

EDIT: so on the numerator i'll have a pmf for a lambda(t-t1) and the denominator the pmf for a lambda(t) and I go from there to get the answer?
 
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#9
You need to know some LaTeX syntax, and put the code inside the math tags.



You miss something?
Yeah think i'm missing something, just not quite sure what, i'm thinking either poisson(lambda(t)) or poisson(lambda(t1) but maybe I'm just not getting something.

EDIT: the numerator of the conditional probability is asking me whats the probability of m arrivals between 0 and t1 multiplied by n arrivals between 0 and t, which i'm thinking is m+n arrivals between t1 and t, and as I discovered this is poisson(lambda(t-t1)). the denominator is simply asking the probability of n arrivals from 0 to t1, so what would I be missing?
 
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BGM

TS Contributor
#10
the numerator of the conditional probability is asking me whats the probability of m arrivals between 0 and t1 multiplied by n arrivals between 0 and t
You mean the multiplication/intersection of the two events right?

which i'm thinking is m+n arrivals between t1 and t
Why m + n??? The question already stated \( t > t_1 \) right?

In order to use the independent property to simplify the joint, please count the process in each non-overlapping interval.
 

Dason

Ambassador to the humans
#11
For the math notation you would write stuff like this:
[noparse]
\(X_t - X_s\)
[/noparse]

which would result in \(X_t - X_s\)
 
#12
You mean the multiplication/intersection of the two events right?



Why m + n??? The question already stated \( t > t_1 \) right?




In order to use the independent property to simplify the joint, please count the process in each non-overlapping interval.

I still can't get this answer, are arrival times uniformally distributed on [0,t1] given Xt=n?