# A question about Poisson process (waiting online)

1. Jul 19, 2015

### ssyldy

Hey guys, I encounter a question (maybe a silly one )that puzzles me. Nt is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]t=Nt, and Nt2-[N,N]t is a martingale, it follows that E[Nt2]=E[[N,N]t]=λ*t. On the other hand, the direct calculation of E[Nt2] can be found in https://proofwiki.org/wiki/Variance_of_Poisson_Distribution, which indicates E[Nt2]=(t*λ)2+t*λ. The two results are different. I really appreciate it if somebody can help me.

Last edited: Jul 19, 2015
2. Jul 19, 2015

### Staff: Mentor

It would help to know the context of that calculation and the meaning of the individual symbols.

3. Jul 19, 2015

### ssyldy

Nt is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]t=Nt, and Nt2-[N,N]t is a martingale, it follows that E[Nt2]=E[[N,N]t]=λ*t. On the other hand, the direct calculation of E[Nt2] can be found in https://proofwiki.org/wiki/Variance_of_Poisson_Distribution

4. Jul 19, 2015

### mathman

It looks like someone has confused the second moment with the variance.

5. Jul 19, 2015

### ssyldy

Hi dude, seems like you know the answer. Could you explain?

6. Jul 20, 2015

### mathman

General formulas for a random variable X:
second moment: $E(X^2)$
variance: $E(X^2)-(E(X))^2$

7. Jul 20, 2015

### ssyldy

Yeah, I know that. But my question is, why would I get two different results of E[Nt2] using two different methods?

8. Jul 21, 2015

### mathman

My guess - a mistake in the derivation. $E[N_t^2]=\lambda t$ looks wrong (unless the mean=0). It is the variance.