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I'm doing a research and using random variables to model a random process. I'm defining a Bernoulli random variable as a product of several other Bernoulli variables (three or more). So, I have the following as the expectation:

$E[X_1]=E[X_2X_3X_4]$

I cannot assume the independence of the variables $X_i$ for any $i$. The issue is to find a correct formula for the expectation to consider the correlation of the variables $X_2$ to $X_4$. I know that for a product of two variables, I can have: $E[X_2X_3]=E[X_2]E[X_3] + Cov(X_2,X_3)$. But this covariance will be a matrix in case of three or more variables. Then, how would one compute a value for $E[X_1]$?

I appreciate if somebody can help with this or point me to a good resource (e.g. book).

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Well, $X_1 = 1$ only when $X_2 = X_3 = X_4 = 1$ and is $0$ otherwise, therefore

$$E(X_1) = P(X_2 = 1, X_3 = 1, X_4 = 1)$$

As @leonbloy mentions, knowledge of the correlations and marginal success probabilities is not sufficient for calculating $E(X_1)$, but it can be written in terms of the conditional probabilities; using the definition of conditional probability,

$$ E(X_1) = P(X_2 = 1, X_3 = 1 | X_4 = 1) \cdot P(X_4 = 1) $$

and $P(X_2 = 1, X_3 = 1 | X_4 = 1)$ can be similarly decomposed as

$$ P(X_2 = 1 | X_3 = 1, X_4 = 1) \cdot P(X_3 = 1 | X_4 = 1) $$

implying

$$E(X_1) = P(X_2 = 1 | X_3 =1, X_4 = 1) \cdot P(X_3 =1 | X_4 = 1) \cdot P(X_4 = 1)$$

Explicit calculation of $E(X_1)$ will require more information about the joint distribution of $(X_2,X_3,X_4)$. The above expression makes sense intuitively - the probability that three dependent Bernoulli trials are successes is the probability that the first is a success, and the second one is a success given the first, and the third is a success given that the first two are. You could equivalently interchange the roles of $X_2, X_3, X_4$.

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  • $\begingroup$ I see what you mean. Since we have $E[X_1]=P(X_1=1)$, we can use conditional probabilities to define $E[X_1]$. In fact, I'm trying to form a general case not limited to a product of three variables; $X = \prod X_i$ for some $i$. I was hopping to find a formulation in terms of a covariance/correlation. But I think I stick to what you suggested. Thanks! $\endgroup$
    – hsnm
    May 27, 2012 at 16:36
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From your definition, the expectation of $X_1$ is given by third cross moment of the variables $X_2,X_3,X_4$, which, in general is not reducible to their correlations (second moment), unless you put some other conditions (or unless I've missed something).

Regarding general formulation of a multivariate Bernoulli variable, perhaps you find this answer useful.

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  • $\begingroup$ +1. Yes but it is expressible in terms of the conditional correlations, knowledge of which would require more information about the association structure. $\endgroup$
    – Macro
    May 26, 2012 at 18:10
  • $\begingroup$ This statement is true, but it takes a little work to show, because the situation is not a general one. For instance, the entire joint distribution of three Bernoullis is determined by just eight probabilities. They sum to unity and knowing their expectations and correlations gives six more linearly independent pieces of information: seven pieces of information in all. Your assertion amounts to saying that no single probability can be uniquely determined from these seven pieces of information. $\endgroup$
    – whuber ♦
    May 26, 2012 at 20:30
  • $\begingroup$ @whuber I'm giving an example in the description. In fact, I need a general case. $\endgroup$
    – hsnm
    May 27, 2012 at 16:37
  • $\begingroup$ @leonbloy What do you mean by some other conditions? Can you elaborate more? Thanks. $\endgroup$
    – hsnm
    May 27, 2012 at 16:38

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