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 April 16th, 2018, 10:39 AM #1 Newbie   Joined: Apr 2018 From: Germany Posts: 3 Thanks: 0 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
 April 16th, 2018, 01:25 PM #2 Global Moderator   Joined: May 2007 Posts: 6,612 Thanks: 617 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.
 April 17th, 2018, 01:15 AM #3 Newbie   Joined: Apr 2018 From: Germany Posts: 3 Thanks: 0 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
April 17th, 2018, 12:56 PM   #4
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 Originally Posted by consuli 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.

 April 18th, 2018, 05:31 AM #5 Newbie   Joined: Apr 2018 From: Germany Posts: 3 Thanks: 0 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
April 18th, 2018, 04:03 PM   #6
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 Originally Posted by consuli 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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