Uniform Distribution and Gaussian Distribution

[rain_ee56]

Hi All,

How can we determine whether to use Uniform Distribution or Gaussian Distribution when solving a problem?
Is there any criteria in choosing the distribution function?

Thank you very much for ur help.

 

[klystron]

Please give us more information on the problem / application.
The normal and Gaussian distributions are the same

 

[rain_ee56]

Thanks klystron for ur reply. I have corrected my post which is Uniform Distribution and Gaussian Distribution.
Let me illustrate the scenario:

Case A:

If I want to measure the transmitted signal [ S ] from n channels and S is said to be i.i.d Gaussian random process. Why?

Case B:

I have particle which can move freely (with certain velocity, v) in the search space to find the maximum point.
The search space is bounded by upper and lower boundaries, B_up and B_low respectively.
Therefore the range of the search space is : R = | B_up - B_low |
the particle's velocity is initialized randomly in which v~U(-R,R)

Why the velocity is uniform distribution? Based on my understanding, in uniform dist. the probability of occurrence of each event is the same. But in this case the velocity is randomly generated.

 

[klystron]

I do not understand your Case A.
In Gaussian random processes there is a mean value, a standard deviation value and the probability of outcomes can be calculated (likely outcomes above a value, likely outcomes between value....)
In uniform distributions there is no mean value and the likely outcomes between two boundary values are all the same. It is a mathematical concept like the Dirac function that has a uniform spectral distribution

 

[agreatstar]

Gaussian distribution is used when you have some observational data that boosts the probability of a certain event.
In case A, we already know that signal transmitted is say of amplitude 'A'. At receiving end we have a high probability of getting a signal of amplitude 'A' and the probability of signal with value being greater or lesser than a simply goes on decreasing(Signal distortion is caused by noise and has been practically proven that high amplitude noise is rare ie has low probability).
In case B we know particle can have any value in range -v to v. Hence we use uniform distribution. If we would have known that particle was fired with a velocity 'x', we would have considered gaussian distribution, cause we know particle still having velocity 'x' has high probability(It having a different value is also possible because of some interaction with other objects but that case will have low probability).

---------- Post added at 03:15 ---------- Previous post was at 03:14 ----------

Gaussian distribution is used when you have some observational data that boosts the probability of a certain event.
In case A, we already know that signal transmitted is say of amplitude 'A'. At receiving end we have a high probability of getting a signal of amplitude 'A' and the probability of signal with value being greater or lesser than a simply goes on decreasing(Signal distortion is caused by noise and has been practically proven that high amplitude noise is rare ie has low probability).
In case B we know particle can have any value in range -v to v. Hence we use uniform distribution. If we would have known that particle was fired with a velocity 'x', we would have considered gaussian distribution, cause we know particle still having velocity 'x' has high probability(It having a different value is also possible because of some interaction with other objects but that case will have low probability).

 

[Max Planck]

It is very good question. Usually to decide which distribution should be used we must conduct an experiment (in fact, scientists did it for us). Many phenomenons has a Gaussian distribution because of Central Limit Theory, i.e. given phenomenon (e.g thermal noise) is a compound of many principle phenomenons (e.g. the free movement of carriers).

---------- Post added at 10:14 ---------- Previous post was at 10:11 ----------

In uniform distributions there is no mean value

Obviously it is not the truth. UD has a mean which equals (a + b) / 2, where a and b are boundary values.

 

[agreatstar]

Uniform distribution is used when all sample points are equiprobable. Like in case B, the particle can be at distance 2 or 3 or 5 or anything inside the boundary. Any distance is equally possible. Hence we use uniform distribution. Now in case A as I m sending signal S(lets say binary '1') through channel, there is a very high probability that it will be received as binary '1'(although it will corrupt it ut still most probably it will be above the threshold) only and a very little probability that noise will change the signal from '1' to '0'. Hence a signal being transmitted as '1' has a high probab of being received as it is and low prob of being garbled. In such case we use gaussian prob with the mean centered at most probable event.

 

[phongphanp]

Good introduction in the book "The introduction to probability by Willaim Feller" and you should find to read nowadays book also.