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we have skew normal distribution with location=0, scale =1 and shape =0 then it is same as standard normal distribution with mean 0 and variance 1.but if we change the shape parameter say shape=5 then mean and variance also can we fix mean and variance with different values of shape parameter. if we have 3 equation of mean,variance and skewness then how can we fix location, scale and shape parameter.can u explain about it?

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up vote 2 down vote accepted

@Ross Millikan is right: you should look after and understand the background. But as I have answered this question on stackoverflow, let me answer it here also.

Just look after how the mean and variance of a skew normal distribution can be computed and you got the answer! Knowing that the mean looks like:

alt text    and    alt text

You can see, that with a xi=0 (location), omega=1 (scale) and alpha=0 (shape) you really get a standard normal distribution (with mean=0, standard deviation=1):

alt text

If you only change the alpha (shape) to 5, you can except the mean will differ a lot, and will be positive. If you want to hold the mean around zero with a higher alpha (shape), you will have to decrease other parameters, e.g.: the omega (scale). The most obvious solution could be to set it to zero instead of 1. See: alt text

Mean is set, we have to get a variance equal to zero with a omega set to zero and shape set to 5. The formula is known:

alt text

With our known parameters:

alt text

Which is insane :) That cannot be done this way. You may also go back and alter the value of xi instead of omega to get a mean equal to zero. But that way you might first compute the only possible value of omega with the formula of variance given.

alt text

Then the omega should be around 1.605681 (negative or positive).

Getting back to mean:

alt text

So, with the following parameters you should get a distribution you was intended to:

location = 1.256269 (negative or positive), scale = 1.605681 (negative or positive - the opposite sign of location) and shape = 5.

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By this way you will get a distribution with a skewness atound 0.850965 as computed at your corresponding question at stackoverflow (look for the link above). – daroczig Jan 13 '11 at 11:59

If you follow the notation in the Wikipedia article you have the mean is $\xi + \omega \delta \sqrt{\frac{2}{\pi}}$ and the variance is $\omega^2(1-\frac{2\delta^2}{\pi})$ where $\delta=\frac{\alpha}{\sqrt{1+\alpha^2}}$. So if you pick your $\alpha$ you can calculate $\delta$. Your desired variance gives you $\omega$ and finally your desired mean gives $\xi$

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how can we draw skew normal distribution with this location and scale and skewness parameter? – user5620 Jan 12 '11 at 6:41
The formula for f(x) given in the article is exactly what you want. Unfortunately, you need to be able to calculate the error function to use it. Many spreadsheets and computer languages have a command for that. If you have to roll your own, there are expansions given on page 299 of – Ross Millikan Jan 12 '11 at 13:46

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