San José State University Department of Economics |
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of Random Variables Compared to that of the Corresponding Unary Function |
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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.
For small values of the variances of u and v
For simplicity let α=[f(E{u},E{v})−E{f(u,v)}].
Then
Since E{u-E{u}}=0 and E{v-E{v}}=0 the taking of the expected values in the above equation yields
If u and v are independent then Cov(u,v)=0. The above equation then reduces to
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
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
The ratio of the standard deviations is then 1/√2, approximately 0.7.
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