Linearity of expectation ($\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y]$)

For integrable random variables $X_1,\dots ,X_n$ on the same probability space and fixed real numbers $a_1,\dots ,a_n$, linearity of expectation statesThe random variables can be dependent random variables. The formula follows by the linearity of integration; a finite linear combination of integrable random variables remains integrable. For example, if $X$ takes $-1,2,8$ with probabilities $1/2,3/10,1/5$, then $X$ and $2X$ are dependent random variables, yet $\mathbb{E}[X+2X]=3\mathbb{E}[X]=5.1$.

An especially useful application counts how many events occur. Put $I_i=1_{A_i}$, an indicator random variable, and $N={\sum }_i{I}_i$. Since $\mathbb{E}[I_i]=\mathbb{P}(A_i)$, we get $\mathbb{E}[N]={\sum }_i\mathbb{P}(A_i)$. There is no need to compute the whole probability distribution of $N$, or to assume independence. For a uniform random permutation of $n\ge 1$ objects, each object stays fixed with probability $1/n$, so the fixed point count of a uniform random permutation has expected value one for every $n$. ^[ref].

Linearity of expectation concerns addition and multiplication by fixed scalars. Products instead satisfy $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$ under the hypotheses of expectation of a product of independent random variables. For the example above, $\mathbb{E}[X^2]=14.5$, whereas $(\mathbb{E}[X]{)}^2=2.89$. Applying a nonlinear function before averaging can therefore change the answer substantially; use the law of the unconscious statistician to compute that average.

In $n$ independent trials with category probabilities $p_1,\dots ,p_m$, let $K_n$ count the distinct categories observed. Write it as the sum of indicator random variables for the categories' appearances. Category $j$ is absent with probability $(1-p_j{)}^n$, so linearity of expectation gives the displayed formula. The appearance indicators need not be independent. For equally likely categories the expression becomes $m(1-(1-1/m{)}^n)$; uneven probabilities can substantially delay observation of rare categories.

If random variables $X_i$ satisfy ${\sum }_{i}E|X_i|<\infty$, then ${\sum }_i{X}_i$ converges absolutely almost surely, is integrable, and its expected value equals ${\sum }_{i}E[X_i]$. Indeed, the monotone convergence theorem gives $E[{\sum }_i|X_i|]={\sum }_{i}E|X_i|<\infty$, so that nonnegative sum is finite almost surely. It dominates every finite partial sum in absolute value; the dominated convergence theorem then justifies passing the limit through the expected value. No independence is required.

For example, if $|X_i|\le M{q}^i$ for a fixed $M$ and $0<q<1$, the geometric series bound proves the hypothesis directly. This is useful when repeated observations are multiplied by geometrically shrinking distances or weights.