When to use E[a X] = a E[X] (Expectation Property)

Question

The Question is:

If X(t) = Asin(ωt+θ) where θ and ω are constants and A is a uniformly distributed random variable in (0,1). Calculate ensemble mean and correlation and comment on its stationarity.

Is it right if I find mean as following?

E[ X(t) ] = E[ A sin(ωt+θ) ] = sin(ωt+θ) E[ A ]

P.S. When to use this property of expectation and when not to?

Answer 1

In general, as pointed by @angryavian in the comment section, for a non-random $a$ we have $$E[aX]=aE[X]$$

You also have that the following result holds if $a$ is independent of $X$ (note that this generalizes the previous statement as a non-random $a$ is always independent of $X$)

$$E[aX]=E[a]E[X]$$