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April 16th, 2018, 10:39 AM   #1
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How to calculate the moments of a new pdf?

Hello!

I have developed the PDF (probability density function) of a new distribution.

I want to calculate their moments (mean, sd, skewness, kurtosis).

How can I do this?

Consuli
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April 16th, 2018, 01:25 PM   #2
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There are standard formulas for the moments: $m_n=\int_{-\infty}^{\infty} x^n f(x)dx$, where f(x) is the probability density function.
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April 17th, 2018, 01:15 AM   #3
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Thanks.

Meanwhile I have found the following integrals to calculate the moments:

E(x) = Integral f(x) * x dx

Var(x) = Integral f(x) * (x-E(x))^2 dx

s= Var(x)^0.5

M3= Integral f(x) * ((x-E(x))/s)^3 dx

M4= Integral f(x) * ((x-E(x))/s)^4 dx

Where fx) refers to the pdf of the distribution.

Are these equivalent to your formula?

Consuli
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April 17th, 2018, 12:56 PM   #4
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Quote:
Originally Posted by consuli View Post
Thanks.

Meanwhile I have found the following integrals to calculate the moments:

E(x) = Integral f(x) * x dx

Var(x) = Integral f(x) * (x-E(x))^2 dx

s= Var(x)^0.5

M3= Integral f(x) * ((x-E(x))/s)^3 dx

M4= Integral f(x) * ((x-E(x))/s)^4 dx

Where fx) refers to the pdf of the distribution.

Are these equivalent to your formula?

Consuli
These are similar. Your formulas for M3 and M4 are centralized and normalized to correspond to a scale change to zero mean and unit deviation.
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April 18th, 2018, 05:31 AM   #5
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To get this on to point.

My formulas would work out for any distribution, but yours only for standardized distribution, like the standard normal distribution?

Sorry for the little inconvenience, but I need to know in detail, simply.

Consuli
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April 18th, 2018, 04:03 PM   #6
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Quote:
Originally Posted by consuli View Post
To get this on to point.

My formulas would work out for any distribution, but yours only for standardized distribution, like the standard normal distribution?

Sorry for the little inconvenience, but I need to know in detail, simply.

Consuli
My formula is for any distribution defined by a density function f(x). If there is no density function and you have only a distribution function F(x) then you must use a Stieltjes integral. $m_n=\int_{-\infty}^{\infty}x^ndF(x)$.
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