Suppose that we are given two probability density functions.
f(x) with mean 0 and variance sf2
g(x) with mean 0 and variance sg2
Let h(x) = N sqrt( f(x) g(x) ),
where N is the normalization constant.
Can we express variance of h(x) by means of sf2 and sg2 in any other way than direct integration (suppose we don't know how to integrate h(x) )?
Could we tell anything in a more general case h(x) = N h(f(x),g(x))?
variance of product of two pdfs !! kindly restate your question. Not clear!
Let f(x) and g(x) be two given PDFs such that
int_R x^2 f(x) = sf2,
int_R x^2 g(x) = sg2.
Let h(x) be defined as
h(x) = N H( f(x) g(x) ),
normalized so that int_R h(x) = 1,
with H(x,y) being a function of two variables.
[ In my problem H(x,y) = sqrt(xy). ]
I want to find
sh2 = int_R x^2 h(x)
in terms of sf2, sg2 and possibly other moments of f(x), g(x).
Yes, but the result can be also expressed by other moments of f and g, if necessary. Is there any theorem that solves this problem?
It is in fact a good idea to use attachments for maths!
I don't know which is the probabilistic interpretation of the density function h. If there is any, it may help in answering you question.
The first thing that comes to my mind is to use the Cauchy-Schwarz inequality in order to bound the variance of h with the product of the variances of f and g.
Thank you for the hint!
What I am actually trying to do is to combine distributions f(x;K) ang g(x;K) into a third one h(x;K) which is approximately equal to their geometric mean (an observation), and (rigorously) h=f for K=1 and h=g for K=infty.
C-S inequality states that sh2
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