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A caution (or two) regarding mathematical expectation

Published online by Cambridge University Press:  20 June 2025

Leonard M. Wapner
Leonard M. Wapner*
Affiliation:
Division of Mathematical Sciences, El Camino College, Torrance, CA 90506, USA e-mail: lwapner@elcamino.edu
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Extract

Computing mathematical expectation for an experiment involving a finite number of numerical outcomes is straightforward. Let X denote the random variable having n possible values x1, x2, x3,…, xn. Letting pk denote the probability of xk, the expected value of X is

$$E\,(X)\, = \,\sum\limits_{k\, = \,1}^n {{x_k}{p_k}} ,$$

which can be interpreted as a weighted average of all xk, where the weight of each outcome is represented by its probability. But caution is required when interpreting the sum if there are infinitely many outcomes and the series fails to converge absolutely.

Type
Articles
Information
The Mathematical Gazette , Volume 109 , Issue 575 , July 2025 , pp. 247 - 252
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Copyright
© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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