Proof

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Discrete Case

Let f(x) be the probability function such that \(f(x_{i})=P(X=x_{i})\).

\(E[a]=\sum_{i}\ a\cdot f(x_{i})=a\sum_{i}f(x_{i})=a\ \ \ (\ \text{∵}\ \sum_{i} f(x_{i})=1)\)

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Continuous Case

Let f(x) be the probability function such that \(f(x)=P(X=x)\).

\(E[a]=\int_{-\infty}^{\infty}\ a\cdot f(x)\ dx=a\int_{-\infty}^{\infty}\ f(x)\ dx=a\ \ \ (\ \text{∵}\ \int_{-\infty}^{\infty}\ f(x)\ dx=1)\)