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The Relative Variance of a Binary Function
of Random Variables Compared to that of the Corresponding Unary Function

Measured characteristics of males and females have distributions which are at least approximately
normal distibutions. Generally the means of these distribution are not significantly different but
the variances of the distributions for females is significantly less than those for males.
Rodolfo
A. Gonzalez has hypothesized that the genes for such characteristics are carried only on the
x chromosomes. Females have two X chromosomes, one from each parent, and males only one and that one
from his mother. According to Gonzalez' formulation
females would have the average of the abilities encoded in their two chromosomes. This reduces the
variance compared to that of males. It would make the variance for females equal to one half of the
variance for males. The relationship of a female's ability to that of her two parents may be more
complicated than a simple average. This is the investigation of the mathematics of the matter.

Let f(u,v) be a function of two random variables, u and v, which have the same distribution. This
means that the expected value of u and v are the same, E{u}=E{v}, and their variances, Var(u) and
Var(v) are the same. The
function f(u,v) is symmetric; i.e., f(v,u)=f(u,v). Let g(u)=f(u,u) be the corresponding unary
function.

If u and v are independent then Cov(u,v)=0. The above equation then reduces to

Var(f) = α² + f²_{u}Var(u) + f²_{v}Var(v)

Let Var(u) and Var(v) be denoted as σ². Since E{u}=E{v} and f is symmetric
f_{u}(E{u}) is equal to f_{v}(E{v}). Let the common value be denoted as φ.
As σ² → 0, α → 0, so take α to be zero. The above equation
is then

Var(f) = 2φ²σ²

On the other hand if u and v are the same, as in g(u)=f(u,u), then Cov(u,v)=σ².
This means that