Moments in Statistics and Physics

Center of gravity C in physics is where sum of all moments which is mass x distance (md) is zero

$\int m_{i} d_{m_{i}-c}=0$ where m s are masses in the system

Moment of inertia which is more when the masses are farther from C is $\int m_{i} d_{m_{i}-c}^{2}=0$

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In statistics we call $\mu =\sum \frac{n_{i} x_{i}}{N}=\sum \frac{n_{i} }{N}x_{i}=\sum f(x_{i})x_{i}=(when n is \infty)= \int f(x_{i})x_{i}$

and we know that $\int f(x_{i}) (x_{i}}-\mu )=\int f(x_{i}) (x_{i}})-\int f(x_{i}) (\mu )=\mu-\mu\int f(x_{i})=\mu-\mu=0$

Therefore mean is like center of gravity.

and f(x)x is called the first moment of each x, and sum of the moments around mean is zero.

As sum of the moments of masses composing a matter around the center of gravity is zero.

We define $Variance =\sum \frac{n_i(x_i-\mu)^2}{N}=\textup{When N goes to infinity }=\int f(x_{i}) (x_{i}-\mu )^2= \textup{second moment}$

For a variable such as Z for which mu is zero

The second moment is $\int f(Z)Z^2=E(Z^2)$ is not Zero and is more when the observations are more dispersed similar to Moment of inertia which is more when the masses are farther from Center of gravity.

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Consider function M(t)

$M(t) = \int e^{tx}f(x) dx$

since

$ e^{t\,X} = 1 + t\,X + \frac{t^2\,X^2}{2!} + \frac{t^3\,X^3}{3!} + \cdots +\frac{t^n\,X^n}{n!} + \cdots. $

$M(t)=\int (1+tx+\frac{t^2x^2}{2!}+\frac{t^3}{3!}...)f(x) dx$

M(t)=\int (1+tx+\frac{t^2x^2}{2!}+\frac{t^3}{3!}…)f(x) dx

$M(t)=\int (x+\frac{tx^2}{2!}+\frac{t^2x^3}{2}...)f(x) dx$

$M'(0)=\int xf(x)dx=\mu$

$\frac{\partial^2 }{\partial t^2}M={M}''(t)=\int (x^2+tx^3+...)f(x)dx$

${M}''(0)=\int x^2f(x)dx=E(x^2) \textup{ if } \mu=0=\sigma ^2$

Therefore we call M(t) moment generating function 🙂

https://onlinecourses.science.psu.edu/stat414/node/73