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Wald's equation
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(Theorem)
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Let $X_1, X_2,\ldots, X_N$ be a sequence of $N$ iid random variables distributed as random variable $X$ such that
- $N>0$ is itself a random variable (integer-valued),
- the expectation of $X$ $\operatorname{E}[X]<\infty$ and
- $\operatorname{E}[N]<\infty$
Then $$\operatorname{E}\Big[\sum_{i=1}^{N}X_i\Big]=\operatorname{E}[N]\operatorname{E}[X].$$
The integer $N$ from above can be viewed as a stopping time for the stochastic process $\lbrace X_i \mid i\in\mathbb{Z}^+ \rbrace$
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"Wald's equation" is owned by CWoo.
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Cross-references: stochastic process, stopping time, integer, expectation, random variables, iid, sequence
This is version 5 of Wald's equation, born on 2004-10-01, modified 2006-09-16.
Object id is 6267, canonical name is WaldsEquation.
Accessed 3460 times total.
Classification:
| AMS MSC: | 60G40 (Probability theory and stochastic processes :: Stochastic processes :: Stopping times; optimal stopping problems; gambling theory) | | | 60K05 (Probability theory and stochastic processes :: Special processes :: Renewal theory) |
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Pending Errata and Addenda
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