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The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
The tower rule may refer to one of two rules in mathematics: Law of total expectation, in probability and stochastic theory; a rule governing the degree of a field extension of a field extension in field theory
This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy. The definition of E ( X ∣ H ) {\displaystyle \operatorname {E} (X\mid {\mathcal {H}})} may resemble that of E ( X ∣ H ) {\displaystyle \operatorname {E} (X\mid H)} for an event H {\displaystyle H} but these ...
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment. [1] [2] It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). [3]
Also confidence coefficient. A number indicating the probability that the confidence interval (range) captures the true population mean. For example, a confidence interval with a 95% confidence level has a 95% chance of capturing the population mean. Technically, this means that, if the experiment were repeated many times, 95% of the CIs computed at this level would contain the true population ...
The emergence of probability : a philosophical study of early ideas about probability, induction and statistical inference. Cambridge New York: Cambridge University Press. ISBN 978-0521685573. Paul Humphreys, ed. (1994) Patrick Suppes: Scientific Philosopher, Synthese Library, Springer-Verlag. Vol. 1: Probability and Probabilistic Causality.
For an approximately normal data set, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%. Shown percentages are rounded theoretical probabilities intended only to approximate the empirical ...
Then the first, "unexplained" term on the right-hand side of the above formula is the weighted average of the variances, hσ h 2 + (1 − h)σ t 2, and the second, "explained" term is the variance of the distribution that gives μ h with probability h and gives μ t with probability 1 − h.