Dear Yehya NASSER!

Gaussian distribution has fast-falling shape at it's ends, and thus there is very weak difference between truncated and original Gaussian. For example, if you don't truncate distribution you can achieve non-physical (for some systems) negative power, however the probability to achieve it is very small.

To my mind, for power-measurements statistical analysis it is better to use chi-square distribution rather then Gaussian, because chi-square distribution don't generate negative random numbers. For example, if power obtained as J2R, where R is resistance and J is current (which could be negative), then by assuming that J has Gaussian distribution, we obtain that J2 has chi-square distribution with one degree of freedom.

https://en.wikipedia.org/wiki/Chi-squared_distribution

Sincerely,

Daniil D. Stupin

Thanks Dear Daniil D. Stupin:

But with with this distribution I cannot use any of these information like the mean, min ,max and the standard deviation.

So what I am searching is a pdf thaf fits the bell shape with the usage of the available information I have.

Many Thanks,

Yehya

Strictly speaking, if the distribution has rigorous, non-infinite, upper and lower bounds, it is not Gaussian, it is something else. That said, I think the idea here is that you believe it should closely approximate a Gaussian, but has finite limits on the tails.

There are two obvious ways to deal with the non-physical tails:

1) Simply clip off the tails so that you define your distribution f(x) to be Gaussian in some interval [a

Dear Yehya NASSER!

If you want to fit power data to Gaussian distribution bell-shape

exp[(P-P0)2/(2*sigma2)]/sqrt(2 pi sigma2)

you can use follows methods:

1) If you have only presented in your question four quantities you can set P0=Mean(Power), and sigma= Sdev(power). In this method, so called method of moments, there is no influence of the maximum and minimum values.

https://en.wikipedia.org/wiki/Method_of_moments_(statistics)

Please note, that it will be not good distribution estimation.

2) If you have a set of power values, then you can build a histogram and fit it to Gaussian by non-linear least squares algorithms.

This method has implementation in MatLab

https://www.mathworks.com/help/stats/model-data-using-the-distribution-fitting-tool.html

3) If you again have set of power values and your finally goal is estimation of the mean value, you can use classical solution of this problem, namely mean estimation based on the Student's or t-distribution.

https://en.wikipedia.org/wiki/Student%27s_t-distribution

https://www.youtube.com/watch?v=HuaojGVJkcU

https://www.youtube.com/watch?v=jFOyzEJctUU

You can also check your data by normality tests to

be certain, that you power-values can be modeled by Gaussian distribution.

https://en.wikipedia.org/wiki/Normality_test

Do I correctly understand, that Sdev(power)=1.029 mW?

Sincerely,

Daniil D. Stupin

Dear Yehya NASSER!

As you have power values, it is obvious that only positive values are allowed. Therefore, I suggest computing the probability density w(x) with the mean and the standard deviation values obtained from the measured values. For me, min and max values are not significant, because these values depend on the number of measured values. Then, you must integrate w(x) from 0 to infinity and use w*(x) = w(x)/I, where I is the computed integral. Hence, the obtained w*(x) is a probability density similar to the Gaussian probability density w(x) but defined only for x ≥ 0. It will be useful to compare w*(x) with a normalized histogram. For this propose, you must divide the interval [min, max] in M sub-intervals of length L = (max – min)/M. I suggest to use M = round(sqrt(N)), where N is the number of measured values and sqrt(N) means the square root of N. Each sub-interval Jk contains nk power values, so you can obtain a vector n = [n1, n2, … nM]. From w*(x) you can also compute the “theoretical” probability pk in order to obtain power values within the sub-interval Jk. As pk = mk/N, you can obtain the “theoretical” number mk of power values within Jk. In order to compare the histogram with the “theoretical” probability density w*(x), it is sufficient to compare the “measured” vector n = [n1, n2, … nM] with the “theoretical” vector m = [m1, m2, … mM]. For example, you can use the relative error e = 100*norm(n-m)/norm(m) %, where norm(v) represents the norm of the vector v. For example, you can use the Euclidean norm. If e < 10%, the truncated Gaussian probability density w*(x) can be validated.