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For an integrable random variable $Y$, the conditional expectation satisfies $|\mathbb{E}[Y\mid \mathcal{G}]|\le \mathbb{E}[|Y|\mid \mathcal{G}]$ almost surely. Taking expected values gives the L1 contraction of conditional expectation. Applying it to $Y_n-Y$ shows that convergence in L1 is preserved by conditional expectation with respect to any fixed sigma-algebra.

Fix $k\ge 1$, ${\gamma }_0>1/(k+1)$, and ${\gamma }_0\le \gamma \le 1-\omega (n)/\log n$ with $\omega (n)\to \infty$. A binomial random graph has one giant component of the displayed order and all other graph components are tree components of orders at most $k$, with high probability. The expected value and variance of the isolated vertex count concentrate it around $n^{1-\gamma }\to \infty$. A Cayley formula upper bound for connected sets excludes component sizes $k+1$ through a small fixed fraction of $n$; the empty-graph cut bound excludes the remaining sizes up to $n/2$. An extra-edge count excludes cyclic small graph components. The expected value of the number of vertices in small nonisolated tree components is $O(nd{e}^{-2d})=o(n{e}^{-d})$, where $d=np$. The Markov inequality completes the count of vertices outside the unique large graph component.

A loss function assigns a numerical penalty to a prediction and its observed outcome. Its expected value is the population risk optimized by a statistical learning procedure.

The Lubell mass of a set family $\mathcal{A}\subseteq \mathcal{P}([n])$ isIt is the expected value of the number of members of $\mathcal{A}$ on a Uniformly random maximal chain in a Boolean lattice.

For a fixed set $L$ of $\lfloor \sqrt{n}\rfloor$ vertices, let ${\tau }_L$ be the first step at which they all belong to one graph component. In the uniform random graph process, ${\tau }_L$ lies between $n(\log n-\omega )/4$ and $n(\log n+\omega )/4$ with high probability for every $\omega \to \infty$. Below that scale the marked isolated vertex count has divergent expected value and small relative variance. Above it the logarithmic-regime giant component misses $o(\sqrt{n})$ vertices; exchangeability and a union bound show that none are marked. The label coupling then transfers the two assertions to the process.

Among distributions on $\{0,1,2,\dots \}$ with fixed expected value $\mu$, the geometric distribution with success probability $1/(1+\mu )$ uniquely maximizes information entropy. Its entropy is

For estimating the expected value over all continuous probability density functions on $[0,1]$, compare $f=1$ and $g_n(u)=1+(u-1/2)/\sqrt{n}$. Their means differ by $1/(12\sqrt{n})$, and ${\chi }^2(P_{g_n}\Vert P_f)=1/(12n)$. The chi-squared divergence of product measures bounds joint total variation distance by $1/2$. The metric squared-loss two-point bound then gives the displayed uniform minimax risk lower bound.

For a nonnegative random variable with finite expected value and $\mathbb{P}(X>t)>0$, its mean residual life is the remaining conditional expected value after surviving past $t$. The tail integral formula for moments gives $m_X(t)={\int }_t^{\infty }\mathbb{P}(X>x)dx/\mathbb{P}(X>t)$. The exponential distribution has constant mean residual life, equal to its original mean. In excess of loss reinsurance, the total variance stationary condition for excess of loss sets a candidate retention equal to this quantity.

Independent planar unit steps with uniformly distributed direction have zero vector expected value. The squared Euclidean norm of their sum has expectation $n$, because each diagonal term is one and all cross terms vanish by independence. More generally, zero-mean independent steps with common second moment $s^2$ give mean-square displacement $n{s}^2$; nonzero drift adds a quadratic contribution.

The mass-normalized first velocity moment of a distribution on phase space. For a velocity-only distribution, omit the position integral. Nonzero finite mass and a finite first moment are required. For a probability distribution this is its expected value; its unnormalized counterpart is the momentum.

The misclassification risk of a classifier is its expected misclassification loss,

For per-capita birth and death rates $B$ and $D$, the rates are $b_n=Bn$ and $d_n=Dn$. If $\mu =\mathbb{E}{N}_t$ and ${\sigma }^2=\mathrm{var}{N}_t$, thenThe expected value sees only the net growth rate $B-D$, whereas the variance also sees the total event rate $B+D$.

For a positive-support exponential family $f(x\mid \theta )=p(x)e^{-\theta x}/q(\theta )$, normalization gives $m(\theta )=-q^{\prime }(\theta )/q(\theta )$. The natural conjugate prior proportional to $q(\theta {)}^{-k}{e}^{-k\mu \theta }$ has logarithmic derivative $k(m(\theta )-\mu )$. If its endpoint density values vanish, integration gives prior expected value $\mathbb{E}[m(\Theta )]=\mu$. Observations update $k$ to $k+n$ and $\mu$ to $(k\mu +{\sum }_i{x}_i)/(k+n)$. The endpoint control for a Laplace-family conjugate posterior justifies the same mean identity after updating, giving an exact credibility estimate with credibility factor $n/(n+k)$.

In one dimension, a nearest neighbour density estimator is ${\hat{f}}_{(k)}(x)=k/[2(n+1){\rho }_{(k)}(x)]$. It uses the nearest neighbour distance to select an interval containing a fixed number of observations. The factor $n+1$ corresponds to the exact expected value of the associated uniform order statistic.

Nested Monte Carlo approximates an outer expected value whose integrand contains another expected value. A nonlinear transformation, such as a logarithm, of an inner sample average usually introduces finite-sample bias. Increasing both sample sizes requires suitable integrability controls to establish consistency; an unbiased estimator of the inner quantity does not make its logarithm unbiased.

The occupation time records the duration spent in a state of a continuous-time Markov chain. Conditional on $Z(0)=r$, its expected value is ${\int }_0^t{p}_{rs}(u)du$, where $p_{rs}$ is the transition probability.