Let {X(t); t $\ge 0$ } be a Poisson process with rate $\lambda=2$. Determine $E[X(1)X(2)]$.
How to determine the mean of those random variables?
Because I think in this case I cannot do $E[X(1)X(2)]=E[X(1)]E[X(2)]$.
Then what should I do?
2026-03-30 03:53:33.1774842813
Calculating the mean in a Poisson process
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Use independence of increments: $$ \mathbb{E}[X_1X_2]=\mathbb{E}[X_1(X_2-X_1)]+\mathbb{E}[X_1^2]=\mathbb{E}[X_1]\mathbb{E}[X_2-X_1]+\mathbb{E}[X_1^2]$$ The three expectations can now be easily computed using the distributions of $X_1$ and $X_2-X_1$.