Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYand differentiate it repeatedly with respect to α (see section 5.12). Thus, we obtain∫dI∞dα = − y 2 exp(−αy 2 ) dy = − 1 2 π1/2 α −3/2−∞d 2 ∫I ∞dα = y 4 exp(−αy 2 ) dy =( 1 )( 3 2 2 2 )π1/2 α −5/2−∞.d n ∫I∞dα n =(−1)n y 2n exp(−αy 2 ) dy =(−1) n ( 1 )( 3 ) ···( 1 (2n − 2 2 2 1))π1/2 α −(2n+1)/2 .−∞Setting α =1/(2σ 2 ) and substituting the above result into (30.55), we find (for k even)ν k =( 1 )( 3 ) ···( 1 (k − 2 2 2 1))(2σ2 ) k/2 = (1)(3) ···(k − 1)σ k . ◭One may also characterise a probability distribution f(x) using the closelyrelated normalised and dimensionless central momentsγ k ≡ν k= ν kν k/2 σ k .2From this set, γ 3 and γ 4 are more commonly called, respectively, the skewnessand kurtosis of the distribution. The skewness γ 3 of a distribution is zero if it issymmetrical about its mean. If the distribution is skewed to values of x smallerthan the mean then γ 3 < 0. Similarly γ 3 > 0 if the distribution is skewed to highervalues of x.From the above example, we see that the kurtosis of the Gaussian distribution(subsection 30.9.1) is given byγ 4 = ν 4ν22 = 3σ4σ 4 =3.It is therefore common practice to define the excess kurtosis of a distributionas γ 4 − 3. A positive value of the excess kurtosis implies a relatively narrowerpeak and wider wings than the Gaussian distribution with the same mean andvariance. A negative excess kurtosis implies a wider peak and shorter wings.Finally, we note here that one can also describe a probability density functionf(x) in terms of its cumulants, which are again related to the central moments.However, we defer the discussion of cumulants until subsection 30.7.4, since theirdefinition is most easily understood in terms of generating functions.30.6 Functions of random variablesSuppose X is some random variable for which the probability density functionf(x) is known. In many cases, we are more interested in a related random variableY = Y (X), where Y (X) is some function of X. What is the probability density1150