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commish

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Hi,

I want to derive an expression to compute the expected value for the product of three (potentially) dependent RV. In a separate thread, winterfors provided the manipulation at the bottom to arrive at such an expression for two RV.

Does anybody have any guidance on how I can take this a step further to establish a like expression for the product of THREE dependent RV? In other words, given three RV called X, Y, and Z, I want to derive:

[tex]E[X \cdot Y \cdot Z] = ... [/tex]

Thanks!

I want to derive an expression to compute the expected value for the product of three (potentially) dependent RV. In a separate thread, winterfors provided the manipulation at the bottom to arrive at such an expression for two RV.

Does anybody have any guidance on how I can take this a step further to establish a like expression for the product of THREE dependent RV? In other words, given three RV called X, Y, and Z, I want to derive:

[tex]E[X \cdot Y \cdot Z] = ... [/tex]

Thanks!

winterfors said:No need to look at conditional PDFs. We have that:

[tex]

Cov[X,Y] = E[(X-E[X])\cdot(Y-E[Y])] [/tex]

[tex]

= E[X \cdot Y] - E[X \cdot E[Y]] - E[E[X] \cdot Y] + E[E[X] \cdot E[Y]] [/tex]

[tex] = E[X \cdot Y] - E[X] \cdot E[Y] [/tex]

Thus,

[tex]E[X \cdot Y] = Cov[X,Y] + E[X] \cdot E[Y] [/tex]

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