Normalising a velocity spectrum

  • Context: Graduate 
  • Thread starter Thread starter natski
  • Start date Start date
  • Tags Tags
    Spectrum Velocity
Click For Summary

Discussion Overview

The discussion revolves around the normalization condition for a velocity distribution, specifically addressing the relationship between probability and velocity in the context of probability distributions. Participants explore the mathematical definitions and implications of normalizing a velocity spectrum.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant initially assumed that the area under the velocity distribution graph should equal 1, but later recognized that the area has dimensions of velocity, questioning the validity of this assumption.
  • Another participant asserts that the y-axis represents probability and suggests normalizing to 1.
  • A different participant agrees that the y-axis is probability but raises concerns about setting the mean velocity to a probability of 1, suggesting this would lead to a total probability exceeding 1.
  • One participant provides a mathematical expression indicating that the integral of the probability distribution should equal 1, while also relating the integral of velocity times the probability distribution to the mean velocity.
  • A further contribution clarifies the distinction between probability distribution and probability, explaining how the probability density function is defined and its implications for dimensions.
  • Practical applications of probability distributions are discussed, with one participant expressing a preference for cumulative probability in certain contexts due to its clarity compared to probability density.
  • A later reply indicates that the original poster's problem has been resolved, but does not elaborate on the specifics of the resolution.

Areas of Agreement / Disagreement

Participants express differing views on the normalization condition and the interpretation of probability versus probability density. There is no consensus on the best approach or definition, and the discussion remains unresolved regarding the implications of these definitions.

Contextual Notes

Participants highlight the importance of dimensional analysis in the context of probability distributions and the potential confusion arising from different interpretations of probability and probability density. The discussion also touches on practical versus theoretical applications of these concepts.

natski
Messages
262
Reaction score
2
For any given velocity distribution, you have a y-axis with probability and an x-axis of velocity. Without really thinking much about it, I had assumed the normalisation condition was that the area under the graph (the integral of the function w.r.t. velocity) would be equal to 1. Of course, the area under the graph has the dimensions of velocity and so it doesn't make sense to set the area to 1.

So my question is, what IS the normalisation condition for a velocity distribution??
 
Physics news on Phys.org
The y-axis is the PROBABILITY, not the velocity, so normalize to 1.
 
This is what I said? The y-axis is probability.

But I don't think it is right to set the mean velocity to a probability of 1 because now the total probability under the graph will be greater than 1.
 
\int Pdx=1, where P is the probability distribution. P is on the y axis.
\int vPdx=<v>.
 
natski,

The y-axis is the probability distribution (say f), not the probability (say P).
The probability distribution with respect to the velocity v is defined such that

dp = f(v) dv​

gives the probability that the velocity is in the interval [v,v+dv].

The probability that the velocity is between v1 and v2 may be obtained by integration:

[tex]P(v1,v2) = \int_{v1}^{v2} f(v') dv'[/tex]​

This last expression is also a probability, also called cumulated probability.

The dimensions of f(v) are the inverse of the dimensions of a velocity.
The dimensions of P(v1,v2) are those of a simple number, no dimension.

In practical applications, like particle size analysis in the industry or sales statistics or reliability data ..., I prefer to use the cumulated probability. The probability density is not convenient in these practical situations. For example the area under a curve is not really a visible data while the value of the cumulated probability along the y-axis is a clear information. And of course, there is this problem with "strange" units for f . In theoretical physics of course the probability density is more convenient.

Michel
 
Last edited:
Just wanted to say thanks for your help on this. Problem now solved.
 

Similar threads

Undergrad Units of a vector in a velocity vs time graph?
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad An actual meaning of instantaneous velocity
  • · Replies 13 ·
Replies
13
Views
2K
High School Kinematic equations ##\textbf{purely}## from graphs
  • · Replies 49 ·
2
Replies
49
Views
4K
Undergrad Boundary conditions for a stream function in a hydrodynamics problem
  • · Replies 5 ·
Replies
5
Views
1K
Graduate Calculate velocity of streams impinging at differing angles
  • · Replies 30 ·
2
Replies
30
Views
2K
Graduate Boltzmann distribution and the number of molecules with a certain velocity
  • · Replies 10 ·
Replies
10
Views
3K
Graduate Angular Velocity Vector; Relationship to Angular Momentum
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Why are the Navier-Stokes equations inconsistent in this case?
  • · Replies 18 ·
Replies
18
Views
3K
Graduate Velocity Verlet for relativistic simulation
  • · Replies 2 ·
Replies
2
Views
1K
Graduate Path of a projectile with a non-constant gravity
  • · Replies 5 ·
Replies
5
Views
2K