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Draw one random variable $R$, taking values zero and two with equal probabilities. Conditional on $R$, generate a Poisson point process on the real line with constant rate $R$. This is a Cox process directed by the random measure $Rdx$: the same rate is shared by every window. It is a locally finite simple point process and a stationary point process. The law of total expectation gives $\mu (A)=\mathbb{E}[R]|A|=|A|$; the law of total variance and law of total covariance give, for bounded Borel sets $A,B$ with $A\cap B=\varnothing$,The conditional covariance vanishes by independence, but the shared conditional means fluctuate together. This distinguishes conditional from unconditional independence.^[ref]

For a window of length $L$, the exact probability mass function isIn particular, its void probability functional is $(1+e^{-2L})/2$, compared with $e^{-L}$ for the unit-rate Poisson point process. Whole realizations can be empty or comparatively crowded, while their mean density remains one. This example demonstrates shared-rate statistical dependence, rather than a model of short-range offspring clusters.

A sufficient statistic allows the observed likelihood function to be stored through a shorter record, up to a parameter-independent factor. For the Bernoulli sample sum as a sufficient statistic, writeThe second likelihood function belongs to the observed count; the first belongs to the observed sequence. Their absolute values differ by the binomial coefficient, but their likelihood ratios between any two interior parameter values agree. A maximum-likelihood estimator can therefore use just $(n,t)$.

This compression discards which arrangement actually occurred. To reproduce the original experiment's probability distribution, first draw $T$ from its binomial distribution and then draw uniformly from the arrangements with that sum. The second step is independent of $p$, so it supplies no additional evidence about $p$. It does not recover the particular discarded sequence. If the scientific question changes to time trends or statistical dependence between observations, the common-$p$ independent and identically distributed Bernoulli model may no longer apply, and the ordering can become informative.

Two square-integrable random variables $X,Y$ are uncorrelated random variables when $\mathrm{Cov}(X,Y)=0$. Independent random variables with finite second moments are uncorrelated random variables, but zero covariance does not generally imply independence.

For an explicit counterexample, let $X$ take $-1$, $0$, $1$ with equal probabilities, and let $Y=X^2$. Then $\mathbb{E}X=\mathbb{E}{X}^3=0$, so $\mathrm{Cov}(X,Y)=\mathbb{E}{X}^3-\mathbb{E}X\mathbb{E}{X}^2=0$. Nevertheless, $Y$ is completely determined by $X$: $\mathbb{P}(Y=0\mid X=0)=1$, while $\mathbb{P}(Y=0)=1/3$. Thus these are uncorrelated random variables with statistical dependence. In this example, opposite contributions from the odd function $x^3$ cancel; the even function $x^2$ still retains information about $|X|$. The last panel of the covariance diagram shows the relationship.

There are useful special cases. Uncorrelated jointly normal variables are independent, and the Bernoulli independence from zero covariance result supplies another setting where the converse holds. Their distribution assumptions do the extra work; zero covariance alone does not.

Wald's equation turns a random sum into a product of two expected values: the average contribution per step and the average number of steps. Its surprising feature is that the number of steps may depend on what has already happened. A procedure can stop on its first successful Bernoulli trial, for example, while the identity still accounts for its total cost. The crucial restriction is that the decision to take the next step cannot use that step's unobserved outcome.

Let $X_1,X_2,\dots$ be real independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, let $\mu =\mathbb{E}[X_1]$, and put $S_0=0$ and $S_n={\sum }_{i=1}^n{X}_i$. Suppose $N$ is a nonnegative integer-valued stopping time for the natural filtration ${\mathcal{F}}_n=\sigma (X_1,\dots ,X_n)$, with $\mathbb{E}[N]<\infty$. Then $S_N$ is an integrable random variable andThese hypotheses allow statistical dependence between $N$ and the increments; they do not allow access to future increments. The finite-expected value requirement is stronger than stopping almost surely.^[ref]

For the proof idea, count step $i$ only on $\{N\ge i\}$. This event is determined by the first $i-1$ observations, so its indicator function is independent of $X_i$. The Proof of Wald's equation by stopping indicators combines this observation with the tail-sum formula for expectation. The same calculation with $|X_i|$ gives $\mathbb{E}{\sum }_{i=1}^N|X_i|=\mathbb{E}|X_1|\mathbb{E}[N]<\infty$, which justifies interchanging the expected value and the infinite series. This is why the result survives a data-dependent end point without assuming an independent count.

The identity is useful in randomized algorithms and repeated sampling: one can translate an average amount of work into an average number of attempts, or conversely, without first finding the full probability distribution of the total. The examples below separate a permissible stopping rule from a rule that peeks ahead and from a valid rule whose average waiting time is infinite.