Calculating Probability of a Poisson Process w/ Parameter λ

[chimychang]

I need some help on the following question: Let N() be a poisson process with parameter $$\lambda$$.

I need to find that probability that

$$N((1,2]) = 3$$ given $$N((1,3]) > 3$$

I know that this is equal to the probability that

$$P(A \cap B) / P(B)$$ where A = N((1,2]) and B = N((1,3]) > 3, but I'm not sure where to go from there.

 

[haruspex]

$$P(A \cap B) / P(B)$$ where A = N((1,2]) and B = N((1,3]) > 3, but I'm not sure where to go from there.

Yes, that's the right start. Can you write down the value of P(B)?
For P(A&B), you have "N((1,2])=3 and N((1,3]) > 3". Can you translate that into a combination of the event A and some fact concerning N((2,3])?

 

[Ray Vickson]

I need some help on the following question: Let N() be a poisson process with parameter $$\lambda$$.

I need to find that probability that

$$N((1,2]) = 3$$ given $$N((1,3]) > 3$$

I know that this is equal to the probability that

$$P(A \cap B) / P(B)$$ where A = N((1,2]) and B = N((1,3]) > 3, but I'm not sure where to go from there.

Are you sure you have copied the problem correctly? Getting P{N(1,2]=3|N(1,3]>3} is not too difficult (just use the definition and known expressions), but the answer is not particularly enlightening. However, the alternative problem P{N(1,3]>3|N(1,2]=3} gives a much nicer answer, and one that reveals an important property of Poisson processes.