expectation concept
| Definition | The average of a random variable's values weighted by their probabilities, its long-run average over repetitions, also called the mean. It is linear whether or not the variables are independent. Primer S, equations S.4 and S.5. Also expected value. |
|---|---|
| Example | A fair die has expectation (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id expectation, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation S.4. |
| Assumptions and scope | none |
| Prior art | none recorded |
| Evidence | none |
| Reviewed | semantic review 2026-09-09; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation S.4.
\[\mathbb E[X]=\sum_x x\,\Pr[X=x],\qquad \operatorname{Var}(X)=\mathbb E\big[(X-\mu)^{2}\big]=\mathbb E[X^{2}]-\mu^{2},\qquad \sigma=\sqrt{\operatorname{Var}(X)}.\]
Book equation S.5.
\[\mathbb E[aX+bY]=a\,\mathbb E[X]+b\,\mathbb E[Y],\qquad \operatorname{Var}(aX+b)=a^{2}\operatorname{Var}(X),\qquad \operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)\ \text{if independent}.\]
Conditions
none
Ledger
none
First stated
Primer S section S.2 of Data Mining as Observation, added in draft 0.3 (2026-09-09) for the ECE 514 readers whose first courses are far behind. The idea is standard and TSK Appendix C covers it at length.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
none
Used in
Data Mining as Observation primer S, chapters 9.
Related
random variable; variance; mean, median; law of large numbers.
See also
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Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.