- Thread starter mathy1991
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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.

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.

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.

The interval \( (0, t_1] \) and \( (0, t] \) are overlapping intervals and therefore they are dependent. So your statement is wrong.

EDIT: is it poisson(lambda(t-t1))?

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

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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You need to know some LaTeX syntax, and put the code inside the math tags.

You miss something?

You miss 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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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

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

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.

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?