Hi everyone,
I have three random variables i.e X, Y and Z and need to find the expectation of following equation:
E[(X-mean(X)(Y-mean(Y)(Z-mean(Z)] ---------------(1)
I tried to simplify the equation and got something like this
COV(XYZ)+------(other terms), where, COV is the covariance
I, however know the cor-relation matrix between X, Y and Z.
My question is their any way to further simplify and find the value of equation (1)?
Thank you in advance,
Samundra
Hello, you should read something about multiple and partial correlation (covariance), generally - there is no way to compute the cov(X,Y,Z), because the covariance (correlation) is the "measure" of the 2-variable dependency. With 3 and more variables you are getting the covariance (or correlation) matrix between the pairs of X,Y and Z.
See question:
https://www.researchgate.net/post/Is_covariance_of_more_than_two_variables_possible
There is no expression like COV(XYZ).
Covariance is defined only for two variables and denoted as COV(X,Y) - There should be a comma between the variables.
Your equation could be rearranged like the following,
E(XYZ) - E(XY) * mu_z - E(YZ) * mu_x - E(XZ) * mu_y + 2 * mu_x * mu_y * mu_z
where,
* : 'product, multiplicaton operation'
mu_x : mean of X, mean(X), E(X)
mu_y : mean of Y, mean(Y), E(Y)
mu_z : mean of Z, mean(Z), E(Z)
XYZ: a random variable from the multiplication of random variable X, Y, and Z
XY: a random variable from the multiplication of random variable X and Y
YZ: a random variable from the multiplication of random variable Y and Z
XZ: a random variable from the multiplication of random variable X and Z
E(XYZ) = triple integral of xyz*f(x,y,z) from -Inf to +Inf
f(x,y,z): joint density function of random variable X, Y, and Z
E(XY) = double integral of xy*f(x,y) from -inf to +inf
f(x,y): joint density function of random variable x and y
= integral of f(x,y,z) with respect to z from -Inf to +Inf
The result is a scalar value not a vetor nor a matrix.
This may help to you.
Thank you for the responses.
@Kyun-Seop Bae: Does this mean I have to solve E(XYZ) to get result for my expression? I think it'll be extremely difficult as it involves triple integration. Do you have any suggestions?
@Kyun-Seop Bae :: I think there is a typo in equation ...
should be
E(XYZ) - E(XY) * mu_z - E(YZ) * mu_x - E(XZ) * mu_y + 2 * mu_x * mu_y * mu_z
Veronika Bojdova : Right!, Thanks. -> I corrected it now.
Samundra Gurung : If you cannot integrate, I suggest to use simulation.
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