# How does the following equation make sense in case of a Poisson process?

#### user366312

##### New member
$$\displaystyle P(N_1=2, N_4=6) = P(N_1=2, N_4-N_1=4) = P(N_1=2) \cdot P(N_3=4)$$

How does the above equation make sense in case of a Poisson process?

Why is $$\displaystyle (N_3=4)=(N_4−N_1=4)$$?

Can anyone explain?

#### Jomo

##### Elite Member
Can you explain your thoughts? What have you tried/consider?

#### user366312

##### New member
> Can you explain your thoughts? What have you tried/consider?

Thanks for the reply. What I understand is:
1. $$\displaystyle P(N_1=2, N_4=6)$$ means "the probability of count of 2 items arriving at step-1, AND 6 items arriving at step-4.
2. $$\displaystyle P(N_1=2, N_4-N_1=4)$$ means that "the probability of count of 2 items arriving at step-1 AND the 4 items as the difference of steps 4 and 1.
3. which is same as $$\displaystyle P(N_1=2, N_3=4)$$ i.e. "the probability of count of 2 items at step-1 AND 4 items at step-3".

I understand (2) as (1) is represented as a "difference" term in (2).

But, how is (2) and (3) equivalent?

#### Jomo

##### Elite Member
Can you give us the complete problem as it appears in your textbook including the meaning of notations etc?

#### user366312

##### New member

Kindly, see this problem.

#### Jomo

##### Elite Member
Now that you gave us more information I still have the same questions. Where are you stuck and have you tried? Please read our guidlines. We need to know where you are stuck so we know where you need help.