[Math] Approximate probability mass function into normal distribution

MATLABnormal distribution

I have an array of values (mass function) Px and I want to approximate a normal distribution function (in Matlab) from them.

I can plot the mass function using bar(Px) and I would like to plot normal distribution graph given these data, too.

How do I do it (mathematical formual will be enough for me)?

Best Answer

The histfit function in the Statistics toolbox seems to do precisely this: http://www.mathworks.in/help/stats/histfit.html

Related Solutions

[Math] How to find (and plot) a probability distribution function

Try hist(data) to get a histogram.

You can also try

[f,xi] = ksdensity(data)
plot(xi,f)

to get a smoothed histogram.

You can restrict the domain (since the default is the real number line):

support = (0:.01:1)'
[f] = ksdensity(data,support)
plot(support,f)

[Math] How to draw the skewed normal distribution curve

You can use the skew normal distribution with parameters $(ξ,ω,α)$ which can be estimated from the given data. If we set $δ=\dfrac{α}{\sqrt{1+α^2}}$, then the mean, variance and skewness of the skew normal distribution are given by (see the link)

mean: $ξ+ωδ\sqrt{\dfrac2π}$
variance: $ω^2\left(1-\dfrac{2δ^2}{π}\right)$
skewness: $\dfrac{4-\pi}{2} \dfrac{\left(\delta\sqrt{\dfrac2\pi}\right)^3}{ \left(1-\dfrac{2\delta^2}{\pi}\right)^{3/2}}$

Substitute your known values for the mean, variance and skewness to find proper values for the parameters $(ξ,ω,α)$ of the distribution. Approximate values will do (you do not need to solve exactly), because you are based on a sample, and the distribution that you will find does not need to fit exactly to the sample. So, trial and error (with a computer), may help, since this is not an easy to solve system.

Start from the formula of the skewness which depends only on $δ$. That is solve

$$0.3131=\dfrac{4-\pi}{2} \dfrac{\left(\delta\sqrt{\dfrac2\pi}\right)^3}{ \left(1-\dfrac{2\delta^2}{\pi}\right)^{3/2}}$$ to find (approximately $δ$). From $δ$ you can find directly $α$. Now, go to the variance and solve $$0.028=ω^2\left(1-\dfrac{2δ^2}{π}\right)$$ to find $ω$. Use the value for the $δ$ that you already have. Finally, use the formula for the mean (with $ω, δ$ known) to find $ξ$ in a similar way.

Best Answer

Related Solutions

[Math] How to find (and plot) a probability distribution function

[Math] How to draw the skewed normal distribution curve

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