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6.2. Moments of normal distribution

Lecture



We have proved above that the expectation of a random variable subject to the normal law (6.1.1) is equal to   6.2.  Moments of normal distribution and the standard deviation is   6.2.  Moments of normal distribution .

We derive general formulas for central moments of any order.

By definition:

  6.2.  Moments of normal distribution .

Doing a variable change

  6.2.  Moments of normal distribution ,

we will receive:

  6.2.  Moments of normal distribution . (6.2.1)

Applying to the expression (6.2.1) the integration formula in parts:

  6.2.  Moments of normal distribution .

Bearing in mind that the first term inside the brackets is zero, we get:

  6.2.  Moments of normal distribution . (6.2.2)

From the formula (6.2.1) we have the following expression for   6.2.  Moments of normal distribution :

  6.2.  Moments of normal distribution . (6.2.3)

Comparing the right-hand sides of formulas (6.2.2) and (6.2.3), we see that they differ only by the factor   6.2.  Moments of normal distribution ; Consequently,

  6.2.  Moments of normal distribution . (6.2.4)

Formula (6.2.4) is a simple recurrence relation that allows expressing higher order moments in terms of lower order moments. Using this formula and bearing in mind that   6.2.  Moments of normal distribution and   6.2.  Moments of normal distribution , it is possible to calculate the central moments of all orders. Because   6.2.  Moments of normal distribution , then from formula (6.2.4) it follows that all odd moments of the normal distribution are equal to zero. This, however, directly follows from the symmetry of the normal law.

For even   6.2.  Moments of normal distribution The following expressions for consecutive moments are derived from formula (6.2.4):

  6.2.  Moments of normal distribution

etc.

General formula for the moment   6.2.  Moments of normal distribution th order for any even   6.2.  Moments of normal distribution has the form:

  6.2.  Moments of normal distribution ,

where under the symbol   6.2.  Moments of normal distribution understood the product of all odd numbers from 1 to   6.2.  Moments of normal distribution .

So as for normal law   6.2.  Moments of normal distribution , its asymmetry is also zero:

  6.2.  Moments of normal distribution .

From the expression of the fourth moment

  6.2.  Moments of normal distribution

we have:

  6.2.  Moments of normal distribution ,

those. the excess of normal distribution is zero. This is only natural, since the purpose of an excess is to characterize the relative steepness of this law as compared to normal.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis