Here are four very important properties of the expected value:
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\mathbf{E}[c \cdot X]=c \cdot \mathbf{E}[X] and \mathbf{E}[c+X]=c+\mathbf{E}[X], where c is a constant.
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\mathbf{E}[X \cdot Y]=\mathbf{E}[X] \cdot \mathbf{E}[Y], if X \perp Y.
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(Linearity of expectation) \mathbf{E}[X+Y]=\mathbf{E}[X]+\mathbf{E}[Y].
Prove these four properties.