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

========================================================================

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.

===================================================

Consider function M(t)

since

<br /><br /><br /> 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.<br /><br /><br />

 

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

=-=-=-=-=-=-=-=-=-

The  moment generating function (MGF) is equivalent to the two-sided Laplace transform of the probability density function (PDF) evaluated at (-t)

The Laplace transform is a mathematical tool that converts a function of time, f(t), into a function of complex frequency, F(s).

(s) represents the frequency domain, but with a specific twist: it is the complex frequency domain.
While the standard frequency domain (used in Fourier transforms) only measures regular, steady oscillations, the s-domain measures both oscillations and growth or decay.
The variable (s) is a complex number written as:


s=sigma +j*omega
  • Sigma: Represents damping or growth/decay. It dictates how fast a signal dies out or blows up over time.
  • Omega: Represents angular frequency 2*pi*f. It dictates how fast the signal oscillates.

 

 

 
The core reason the Laplace transform (expressed as a Moment Generating Function, or MGF) is useful in probability is that it turns the difficult calculus of combining independent variables into simple algebra, while acting as a factory that manufactures moments (like mean and variance) through basic differentiation.
When you substitute s = -t into the Laplace transform of a probability density function (PDF), you get the MGF: M_X(t) = E[e^{tX}].
Here is exactly why this transformation makes finding the mean and standard deviation so easy:
 
 

1. The Derivative acts as a “Moment Downloader”

In the time domain, finding the mean (E[X]) or the second moment (E[X^2]) requires solving complex integrations:

    \[E[X^n] = \int_{-\infty}^{\infty} x^n f(x) \, dx\]

When you transform to the MGF domain, the Taylor series expansion of e^{tX} structures the function so that every derivative you take drops down a power of X.

    \[M_X(t) = 1 + tE[X] + \frac{t^2}{2!}E[X^2] + \frac{t^3}{3!}E[X^3] + \dots\]

By evaluating the derivative at t = 0, you wipe out all other terms, leaving behind the exact moment you want:
 
  • First Derivative (t=0): Gives you the Mean (\mu = E[X]).
  • Second Derivative (t=0): Gives you E[X^2], which easily unlocks the Standard Deviation via \sigma = \sqrt{E[X^2] - (E[X])^2}.
Instead of integrating over and over, you only have to differentiate.
 
 
 
 
 
 
 

2. Convolution Becomes Simple Multiplication

The most powerful trick of the Laplace/MGF domain is how it handles the sum of independent random variables (Z = X + Y).
 
  • In the Time/Probability Domain: To find the probability distribution of Z, you must solve a highly complex calculus operation called convolution:

f_Z(z) = \int_{-\infty}^{\infty} f_X(x)f_Y(z-x)dx

  • In the Frequency/MGF Domain: Convolution completely unravels. The transformation maps addition in the time domain directly to basic multiplication in the frequency domain:

 
 
 
 
 
 
 
Once you multiply the two transformed functions together, you can immediately find the mean and standard deviation of the combined system by taking the derivatives of the new, multiplied MGF.

3. Summary: Why it Works

[Probability Domain]                         [MGF Frequency Domain]
Hard Integration (Convolution)   =======>   Easy Multiplication (M_X * M_Y)

          |                                            |
     (Hard Math)                                  (Easy Math)
          v                                            v
Hard Structural Integrals        =======>   Simple Derivatives evaluated at t=0
                                            (Yields Mean & Standard Deviation)
By shifting to the frequency domain, you trade unsolvable integrals for straightforward high-school algebra and derivatives.
If you want to see this mechanism in action, let me know:
 
  • Do you have a specific distribution (like the Normal, Exponential, or Binomial) you want to derive the mean for?
  • Would you like to see the step-by-step math for proving the sum of two variables using this trick?
 
=-=-=-=-=-=-=-
Example:
 
The gamma function is a mathematical generalization of the factorial function to real and complex numbers.
In probability theory and statistics, the -distribution with degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

Loading