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In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. It has applications in the analysis of time series. It was introduced by David Brillinger (see References below). Probability theory is the mathematical study of probability. ...
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Statistics is a broad mathematical discipline which studies ways to collect, summarize and draw conclusions from data. ...
// Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ...
Nomenclature in probability theory is not wholly standard. ...
The proposition in probability theory known as the law of total expectation, or the law of iterated expectations, or perhaps by any of a variety of other names, states that if X is an integrable random variable (i. ...
In probability theory, the law of total variance states that if X and Y are random variables on the same probability space, and the variance of X is finite, then In language perhaps better known to statisticians than to probabilists, the first term is the unexplained component of the variance...
In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ...
It is most transparent when stated in its most general form, for joint cumulants, rather than for cumulants of a specified order for just one random variable. In general, we have A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...
 where - κ(X1, ..., Xn) is the joint cumulant of n random variables X1, ..., Xn, and
- the sum is over all partitions π of the set { 1, ..., n } of indices, and
- "B ∈ π" means B runs through the whole list of "blocks" of the partition π, and
- κ(Xi : i ∈ B | Y) is a conditional cumulant given the value of the random variable Y. It is therefore a random variable in its own right—a function of the random variable Y.
A partition of U into 6 blocks: a Venn diagram representation. ...
Examples
The special case of just one random variable and n = 2 or 3 Only in case n = either 2 or 3 is the nth cumulant the same as the nth central moment. The case n = 2 is well-known (see law of total variance). Below is the case n = 3. The notation μ3 means the third central moment. The kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ...
In probability theory, the law of total variance states that if X and Y are random variables on the same probability space, and the variance of X is finite, then In language perhaps better known to statisticians than to probabilists, the first term is the unexplained component of the variance...
General 4th-order joint cumulants For general 4th-order cumulants, the rule gives a sum of 15 terms, as follows:  -
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Cumulants of compound Poisson random variables Suppose Y has a Poisson distribution with expected value 1, and X is the sum of Y independent copies of W. In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...
In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...
 All of the cumulants of the Poisson distribution are equal to each other, and so in this case are equal to 1. Also recall that if random variables W1, ..., Wm are independent, then the nth cumulant is additive:  We will find the 4th cumulant of X. We have:  -
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 (the punch line—see the explanation below). We recognize this last sum as the sum over all partitions of the set { 1, 2, 3, 4 }, of the product over all blocks of the partition, of cumulants of W of order equal to the size of the block. That is precisely the 4th raw moment of W (see cumulant for a more leisurely discussion of this fact). Hence the moments of W are the cumulants of X. This article is in need of attention from an expert on the subject. ...
// Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ...
In this way we see that every moment sequence is also a cumulant sequence (the converse cannot be true, since cumulants of even order ≥ 4 are in some cases negative, and also because the cumulant sequence of the normal distribution is not a moment sequence of any probability distribution). The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ...
Conditioning on a Bernoulli random variable Suppose Y = 1 with probability p and Y = 0 with probability q = 1 − p. Suppose the conditional probability distribution of X given Y is F if Y = 1 and G if Y = 0. Then we have  where  means π is a partition of the set { 1, ..., n } that is finer than the coarsest partition -- the sum is over all partitions except that one. For example, if n = 3, then we have 
References - David Brillinger, "The calculation of cumulants via conditioning", Annals of the Institute of Statistical Mathematics, Vol. 21 (1969), pp. 215-218.
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