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Suppose $X(t)$ and $Y(t)$ be two independent random processes. Is $E(X(t_1)Y(t_2))$ necessarily zero?

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    $\begingroup$ No. $E<X \cdot Y>$ is simply $E<X> \cdot E<Y>$ $\endgroup$ – Hilmar Jul 12 at 10:55
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No. Quoting Wikipedia's article Independence (probability theory):

If $X$ and $Y$ are independent random variables, then the expectation operator $\operatorname{E}$ has the property

$$\operatorname{E}[X Y] = \operatorname{E}[X]\operatorname{E}[Y].$$

Consider your $X(t_1)$ and $Y(t_2)$ as $X$ and $Y$ in this answer. If both $\operatorname{E}[X] \ne 0$ and $\operatorname{E}[X] \ne 0,$ then the product $\operatorname{E}[X]\operatorname{E}[Y]$ will be non-zero.

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