Properties of Expectation for Discrete Random Variables

Establishes the linearity of expectation $E[aX+b]=aE[X]+b$ and the Law of the Unconscious Statistician $E[g(X)]=\sum_x g(x)p(x),$ and combines the two to evaluate expectations of polynomial transformations of a discrete random variable.

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Linearity of Expectation

Let $X$ be a discrete random variable with expected value $E[X],$ and let $a, b \in \mathbb{R}$ be constants. Expectation satisfies the following linearity properties:

$E[b] = b$ — the expectation of a constant is the constant itself.
$E[aX] = aE[X]$ — constants pull out of the expectation.
$E[aX + b] = aE[X] + b$ — the two rules combined.

Each property follows directly from the definition of expectation. For example,

E[aX + b] = \sum_x (ax + b)\, p(x) = a\sum_x x\, p(x) + b\sum_x p(x) = aE[X] + b,

where we used $\sum_x p(x) = 1.$

To illustrate, suppose $X$ has the distribution

\begin{array}{|c|c|c|c|} \hline x & 1 & 2 & 3 \\ \hline P(X=x) & 0.5 & 0.3 & 0.2 \\ \hline \end{array}

Then $E[X] = 1(0.5) + 2(0.3) + 3(0.2) = 1.7,$ so

E[2X + 5] = 2(1.7) + 5 = 8.4.

Notice that we did not need to find the distribution of $2X+5$ — linearity lets us compute the new expectation directly from $E[X].$