What is ρ in the Bivariate Normal distribution?
In going from the the Normal distributions for the horizontal axis, \(\mathcal{N}(\mu_h,\,\sigma_h^2)\), and vertical axis, \(\mathcal{N}(\mu_v,\,\sigma_v^2)\) a new equation was postulated with a parameter \(\rho\).
\( f(h,v; \mu_h, \mu_v, \sigma_h, \sigma_v, \rho) = \frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[ \frac{(h-\mu_h)^2}{\sigma_h^2} + \frac{(v-\mu_v)^2}{\sigma_v^2} - \frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v} \right] \right) \)
First a bit of explanation about what \(\rho\) is. Assuming that two variables are correlated, a simple correlation to propose is that the two variables are linearly correlated. Thus for some point \(i\) the equation of interest is:
\(v_i = v_0 + \beta h_i + \epsilon_i\)
Where \(v_0\) is the intercept along the vertical axis, and \(\beta\) is the slope of the line. Given the locations \((h_i, v_i)\) of the shots in the group on the target, the coefficients \(v_0\) and \(\beta\) are calculated to give a "best" fit to the data. There are two examples of best fits lines shown below. The graph on the right shows the "residuals" from the fit as vertical lines from the horizontal value which is assumed to be accurate to the vertcal value which is assumed to contain the error.