Expected value and linearity: weighted averages without independence

Define the quantity and its units

Let X be the number of support requests arriving during a hypothetical short interval. Assume it can equal zero, one, two or three with probabilities 0.1, 0.2, 0.4 and 0.3. These probabilities are stipulated for teaching and are not measurements of a real service. The OpenStax explanation of expectation gives the probability-weighted sum: multiply each value by its probability and add the products. The units of the result remain requests per interval, even when the numerical answer is not a whole number.

Keep a distribution distinct from a sample

The four probabilities define a model distribution. A sample of ten intervals might produce frequencies that differ from them. The model expectation and the observed sample average answer related but different questions. Penn State’s mathematical expectation lesson distinguishes the mean and other distributional quantities. A sample can help estimate an unknown expectation, but it does not turn an assumed probability table into empirical evidence. Label the origin of every probability: stipulated, estimated from data, or derived from another explicitly described model.

Calculate the weighted average

The result 1.90 is a centre of the distribution, not a forecast that an interval contains a fractional request. The most likely individual value here is two, while the expected value is 1.90. In other distributions those quantities can be much farther apart. Reporting the distribution alongside its mean reveals information that the mean alone cannot preserve.

Use linearity for totals

Suppose X and Y count requests in two successive intervals, each with expectation 1.90. Then E(X+Y)=3.80 whether the intervals are independent or dependent, provided the expectations are well defined. If one busy interval tends to be followed by another, that dependence affects the distribution and variance of the total, but not this addition rule. More generally, E(aX+bY+c)=aE(X)+bE(Y)+c for fixed constants. Keep the constants fixed in the stated calculation; a coefficient that is itself random requires a different treatment.

Do not extend linearity to every operation

In general, E(XY) is not E(X)E(Y), and E(X²) is not the square of E(X). The Stanford probability reference places expectation and variance formulas side by side, making this distinction visible. For the request model, E(X²)=0+0.2+1.6+2.7=4.5, whereas E(X)²=3.61. Their difference, 0.89, is the variance. A nonlinear transformation changes the weighting problem. Compute it from the distribution instead of passing the mean through a square, ratio or reciprocal and assuming the result is unchanged.

Use indicators to simplify a counting problem

An indicator equals one when a specified event occurs and zero otherwise. Its expectation equals the event’s probability. If a document has twenty fields and each field has a stated 0.02 probability of a transcription error, the expected number of erroneous fields is twenty times 0.02, or 0.4, by linearity. This calculation does not require independent errors. However, independence would matter for the probability of at least one error or the distribution of the total. The example illustrates a distinction between an expected count and the chance that a document contains any error.

Distinguish totals from averages with changing denominators

An average computed over a fixed number of observations is a fixed multiple of their sum, so linearity applies directly. A ratio with a random denominator requires more care. For example, the average requests per active interval depends on which intervals qualify as active, and that denominator can vary with the recorded requests themselves. Dividing the expected total by the expected number of active intervals is not automatically the expectation of the ratio. Define the target and how zero qualifying intervals are handled before applying an averaging rule.

Expectation checklist and interpretation

• Name the random quantity and units.
• Check that probability weights sum to one.
• Compute each weighted contribution.
• Distinguish the mean from the most likely value.
• Use linearity for sums with fixed coefficients.
• Check extra assumptions for products and nonlinear transformations.
• Separate a model expectation from observed averages.

When presenting a workload estimate, pair the expectation with information about spread or relevant tail probabilities if those are known. Two services can average the same number of requests while having very different peak loads. A staffing or engineering decision therefore needs more than one summary number. Keep the hypothetical nature of this example explicit: its arithmetic demonstrates a method and does not estimate the workload of a particular organisation.

Sources and verification

• Mean, expected value and standard deviation — OpenStax, Rice University — 2026-10-23
• Mathematical expectation, STAT 414 — Penn State — 2026-10-23
• Probability and statistics refresher — Stanford University teaching materials — 2026-10-23