## An extract on #bestmoments2017

In the limit when {\displaystyle \sigma } tends to zero, the probability density f ( x ) {\displaystyle f(x)} eventually tends to zero at any x {\displaystyle x\neq \mu } , but grows without limit if x = {\displaystyle x=\mu } , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when = 0 {\displaystyle \sigma =0} . However, one can define the normal distribution with zero variance as a generalized function; specifically, as Dirac's "delta function" {\displaystyle \delta } translated by the mean {\displaystyle \mu } , that is f ( x ) = ( x ) . {\displaystyle f(x)=\delta (x-\mu ).} Its CDF is then the Heaviside step function translated by the mean {\displaystyle \mu } , namely F ( x ) = { 0 if x < 1 if x {\displaystyle F(x)={\begin{cases}0&{\text{if }}x<\mu \\1&{\text{if }}x\geq \mu \end{cases}}}

If X1, X2, , Xn are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with n degrees of freedom X 1 2 + + X n 2 n 2 . {\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}\ \sim \ \chi _{n}^{2}.} . If X1, X2, , Xn are independent normally distributed random variables with means and variances 2, then their sample mean is independent from the sample standard deviation, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution with n 1 degrees of freedom: t = X S / n = 1 n ( X 1 + + X n ) 1 n ( n 1 ) [ ( X 1 X ) 2 + + ( X n X ) 2 ] t n 1 . {\displaystyle t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left[(X_{1}-{\overline {X}})^{2}+\cdots +(X_{n}-{\overline {X}})^{2}\right]}}}\ \sim \ t_{n-1}.} ' If X1, , Xn, Y1, , Ym are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with (n, m) degrees of freedom: F = ( X 1 2 + X 2 2 + + X n 2 ) / n ( Y 1 2 + Y 2 2 + + Y m 2 ) / m F n , m . {\displaystyle F={\frac {\left(X_{1}^{2}+X_{2}^{2}+\cdots +X_{n}^{2}\right)/n}{\left(Y_{1}^{2}+Y_{2}^{2}+\cdots +Y_{m}^{2}\right)/m}}\ \sim \ F_{n,\,m}.}

A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size k k {\displaystyle k\times k} , then ( y x ) A ( y x ) + ( x z ) B ( x z ) = ( x c ) ( A + B ) ( x c ) + ( y z ) ( A 1 + B 1 ) 1 ( y z ) {\displaystyle {\begin{aligned}&(\mathbf {y} -\mathbf {x} )'\mathbf {A} (\mathbf {y} -\mathbf {x} )+(\mathbf {x} -\mathbf {z} )'\mathbf {B} (\mathbf {x} -\mathbf {z} )\\={}&(\mathbf {x} -\mathbf {c} )'(\mathbf {A} +\mathbf {B} )(\mathbf {x} -\mathbf {c} )+(\mathbf {y} -\mathbf {z} )'(\mathbf {A} ^{-1}+\mathbf {B} ^{-1})^{-1}(\mathbf {y} -\mathbf {z} )\end{aligned}}} where c = ( A + B ) 1 ( A y + B z ) {\displaystyle \mathbf {c} =(\mathbf {A} +\mathbf {B} )^{-1}(\mathbf {A} \mathbf {y} +\mathbf {B} \mathbf {z} )} Note that the form x A x is called a quadratic form and is a scalar: x A x = i , j a i j x i x j {\displaystyle \mathbf {x} '\mathbf {A} \mathbf {x} =\sum _{i,j}a_{ij}x_{i}x_{j}} In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since x i x j = x j x i {\displaystyle x_{i}x_{j}=x_{j}x_{i}} , only the sum a i j + a j i {\displaystyle a_{ij}+a_{ji}} matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form x A y = y A x . {\displaystyle \mathbf {x} '\mathbf {A} \mathbf {y} =\mathbf {y} '\mathbf {A} \mathbf {x} .}