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My distribution histogram looks like it is not identically distributed, as in the negative counts have a different shape than the positive counts. Here is an image:

enter image description here

The chart has 800 data points, and the tallest count of 40 is for 0.

I don't know a formal approach, but looking at it, it appears to me it is not identically distributed. My guess is that the negative counts (to the left of 0) are closer to a normal distribution, while the positive counts (to the right of 0) are closer to a Laplace distribution, or have higher kurtosis than the left side.

Is this possible, are there cases where this happens, is there a topic in probability that studies it? Or, is this just my eyes playing tricks on me, or not having enough sample data or some other simple reason?

Thanks in advance for any help.

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    I can do that!${}$2012-04-17

2 Answers 2

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"Shape parameters" are used to describe changes, such as "skewness", in a family of distributions other than those generated by "location" or "scale" parameters. There are many examples of distributions with shape parameters. Your distribution actually looks like a product of two normal distributions with different means, like $N(−1,1)⋅N(1,1)$.

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As @cardinal wrote, you probably don't mean "identically distributed": that's something you can't really tell from a histogram. If you want to test whether this is a normal distribution, you might use a D'Agostino-Pearson test, a Shapiro-Wilk test, or a Kolmogorov-Smirnov test. See e.g. http://www.graphpad.com/faq/viewfaq.cfm?faq=959 and references there.

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    Yes, of course it can. The only requirements on a PDF are that it is nonnegative everywhere and its integral over the real line is $1$.2012-04-17