Mean and s.d. of product of two variables

[Monique]

I'm not a math wizard and now I need to calculate the mean and s.d. of the product of two variables. Apparently there is a delta method that should be able to do the trick, but I don't know how to apply it. I have SPSS, should that be able to help me out?

 

[Monique]

I found this: http://www-stat.stanford.edu/~susan/courses/s200/lectures/lect5.pdf

Unfortunately the formulas are a bit too much for me. Does the second formula on page 2 make sense to anyone?

 

[Enuma_Elish]

In general you need to use an approximation method (e.g. Taylor series) or simulation.

A useful formula is: E[XY] = E[X]E[Y] + Cov[X,Y]. This assumes you know the covariance between X and Y.

As for the formulas, Y = g(X) is approximated around the mean of X, m, as g(m) + (X - m)g'(m) + (X - m)^2 g"(m)/2. In a first order approximation the second-order term (X - m)^2 g"(m)/2 is assumed zero, so I can write Y = g(m) + (X - m)g'(m) and therefore E[Y] = E[g(m) + (X - m)g'(m)] = E[g(m)] + E[(X - m)g'(m)]. Since m is a constant, for an arbitrary function h and an arbitrary random variable Z, E[h(m)] = h(m) and E[Z h(m)] = h(m)E[Z]. Hence E[Y] = g(m) + g'(m)E[(X - m)] = g(m) + g'(m)(E[X] - m) = g(m), because E[X] = m.

In a second-order approximation, you have the additional term E[(X - m)^2 g"(m)/2] = g"(m)E[(X - m)^2]/2 = g"(m)Var[X]/2.

EnumaElish
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