Section 8.2

The Normal Distribution

Linear combinations of independent normals, CLT approximations, bivariate-normal orthant probabilities (Sheppard's formula), the aim that maximises a normal hit probability.

Toolkit

Normal families are closed under affine combination: if $X_i\sim\mathcal N(\mu_i,\sigma_i^2)$ independently, then $\sum a_iX_i\sim\mathcal N(\sum a_i\mu_i,\sum a_i^2\sigma_i^2)$ . For centred independent (or jointly) normals, symmetry gives $\mathbb P(\text{linear combo}>0)=1/2$ . CLT: for i.i.d. terms with mean $\mu$ , variance $\sigma^2$ , $\frac{S_n-n\mu}{\sigma\sqrt n}\to\mathcal N(0,1)$ , so $\mathbb P(S_n\le x)\approx\Phi\!\big(\frac{x-n\mu}{\sigma\sqrt n}\big)$ .

Standard bivariate normal with correlation $\rho$ : rotational symmetry (for $\rho=0$ ) makes the probability of a wedge of angle $\theta$ through the origin equal $\theta/2\pi$ ; more generally, Sheppard’s formula gives

\mathbb P(X>0,Y>0)=\frac14+\frac{\arcsin\rho}{2\pi}.

For a small target of width $w$ at point $t$ , hit probability $\approx w\,f(t)$ where $f$ is the aiming density; to split effort among several tiny targets, maximise $\sum_i w_i f(t_i;\mu,\sigma)$ over the aim $\mu$ .

Problems