Moments and their Interpretation
Mathematical TreatmentThis is part of the moments math collection.
Moments are deﬁned for every distribution. These are generalized functions which ﬁnd use in a variety of statistical experimental data analysis. In particular, we brieﬂy elucidate how various moments are used to characterize the shape of a distribution.
Finally, we note in passing that a distribution may be totally described via its moments.
The expectation is called the moment of the random variable about the number
Moments about zero are often referred to as the moments of a random variable or the initial moments.
The moment satisﬁes the relation:
When , then the moment of the random variable about is called the central moment.
The central moment satisﬁes the relation:
Remark: We note that and for random variables.
Central and Initial Moment Relations
Also we note that, for distributions symmetric about the expectation, the existing central moments of even order are zero.
Condition for Unique Determinacy
A probability distribution may be uniquely determined by the moments provided that they all exist and the following condition is satisﬁed:
The absolute moment of about is deﬁned by:
The existence of a moment or implies the existence of the moments and of all orders
We note in passing that the mixed second moment is better known as the covariance of two random variables and is deﬁned as the central moment of order (1+1):
We note the following:
- The ﬁrst initial moment is the expectation.
- The second central moment is the variance.
- The third central moment is related to the skewness.
- The fourth central moment is related to the kurtosis.
A measure of lopsidedness, for symmetric distributions. (a.k.a asymmetry coefﬁcient)
A measure of the heaviness of the tail of the distribution, compared to the normal distribution of the same variance. (a.k.a excess, excess coefﬁcient) Essentially a measure of the tails of the distribution compared with the tails of a Gaussian random variable. (Florescu and Tudor 2013)
Mathematically (Polyanin and Chernoutsan 2010):
Florescu, I., and C.A. Tudor. 2013. Handbook of Probability. Wiley Handbooks in Applied Statistics. Wiley. https://books.google.co.in/books?id=2V3SAQAAQBAJ.
Polyanin, A.D., and A.I. Chernoutsan. 2010. A Concise Handbook of Mathematics, Physics, and Engineering Sciences. CRC Press. https://books.google.co.in/books?id=ejzScufwDRUC.