What is the meaning of expectation, variance and standard deviation?

Question

Mathematical analysis

What is the physical meaning of these common concepts of probability theory and mathematical statistics, which often go together with each other. Everywhere there are solid formulas and dry definitions that do not make it clear what the real meaning is and how it can be applied to a real demonstrative example, and not to a meaningless set of the same type of examples given in textbooks and websites.
As a rule, the following dry formulations are given: " average value of a random variable ", " measure of the spread of a given random variable ", " dispersion index of the values ​​of a random variable ", accompanied by formulas.
Can anyone explain what is the physical meaning of these concepts on a real practical example.

Answer 1

Everywhere there are solid formulas and dry definitions that do not make it clear what the real meaning is and how it can be applied in a real illustrative example.

If you don't like "solid formulas" - then maybe you don't need it?
But here's an example ... for example, for random noise recorded by successive samples in an array (you called yourself a "programmer"?):
- mathematical expectation is the average value over the array samples, or the constant component of the signal;
- dispersion is the power of the signal;
- RMS (standard deviation) is the average amplitude of the signal;
That's the "physical meaning" for you.

Answer 2

There seems to be a logical connection between the spread (dispersion) and the inverse square law (fall in intensity with distance from the source).
https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D...