"Two independent random variables x and y have PDF's given by f(x) = 12x^2 (1 - x) for 0 <= x <= 1 and f(y) = 2y for 0 <= y <= 1. Their product Z = XY has PDF defined as f(z) = 12z(1 - z)^2 for 0 <= y <= 1. Find the approximate values obtained by the method of statistical differentials."(adsbygoogle = window.adsbygoogle || []).push({});

I have already got the answer when I used the formula (not the required in the problem above) for the expected value and the variance of

12z(1 - z)^2...I got 0.4 and 0.04 respectively and the same answer resulted when I solved for E[XY] and var(XY).

But I got crazy in using the method of statistical differentials because I have not encountered this before, but the formula truncates to 3 terms( I have seen the formula)... and uses the taylor series... and I don't quite understand how to solve the problem... the answers, according to the book, for the approximate mean and variance are 0.4 and 0.0377, respectively. What must be my g(x,y) in this problem? Maybe from this, I can have an idea.....

How do I solve this problem?

The book says let m_1 = E[X] and m_2 = E[Y]... the formula for the statistical differentials requires the known values for Var(X) and Var(Y) and Cov(X,Y)...

E[g(X,Y)] = g(m_1, m_2) + ½g"(x)(m_1, m_2)Var(X) + ½g"(y)(m_1, m_2)Var(Y)+ g"(x,y)(m_1, m_2)Cov(X,Y)

where g"(x) is the second partial derivative of g(x,y) wrt x;

g"(y) is the second partial derivative of g(x,y) wrt y;

g"(x,y) is the partial derivative of g(x,y) wrt y, then wrt x;

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# Homework Help: Some help on a tough math topic

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