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P

The normal (Gaussian) distribution has the form

1

PN (x) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€(x Ã€ m)2 =2s2 ] (3:2:9)

2ps

22 Probability Distributions

It is often denoted N(m, s). Skewness and excess kurtosis of the

normal distribution equals zero. The transform z Â¼ (x Ã€ m)=s con-

verts the normal distribution into the standard normal distribution

1

PSN (z) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€z2 =2] (3:2:10)

2p

Note that the probability for the standard normal variate to assume

the value in the interval [0, z] can be used as the definition of the error

function erf(x)

Ã°

z

pï¬ƒï¬ƒï¬ƒ

1

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp (Ã€x2 =2)dx Â¼ 0:5 erf(z= 2) (3:2:11)

2p

0

Then the cumulative distribution function for the standard normal

distribution equals

pï¬ƒï¬ƒï¬ƒ

PrSN (z) Â¼ 0:5[1 Ã¾ erf(z= 2)] (3:2:12)

According to the central limit theorem, the probability density distri-

bution for a sum of N independent random variables with finite

variances and finite means approaches the normal distribution as N

grows to infinity. Due to exponential decay of the normal distribu-

tion, large deviations from its mean rarely appear. The normal distri-

bution plays an extremely important role in all kinds of applications.

The Box-Miller method is often used for modeling the normal distri-

bution with given uniform distribution [4]. Namely, if two numbers

x1 and x2 are drawn from the standard uniform distribution, then

y1 and y2 are the standard normal variates

y1 Â¼ [Ã€2 ln x1 )]1=2 cos (2px2 ), y2 Â¼ [Ã€2 ln x1 )]1=2 sin (2px2 ) (3:2:13)

Mean and variance of the multivariate normal distribution with N

variates can be easily calculated via the univariate means mi and

covariances sij

X X

N N

mi , s2

mN Â¼ Â¼ (3:2:14)

sij

N

iÂ¼1 i, jÂ¼1

The lognormal distribution is a distribution in which the logarithm of a

variate has the normal form

23

Probability Distributions

1

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€( ln x Ã€ m)2 =2s2 ]

PLN (x) Â¼ (3:2:15)

xs 2p

Mean, variance, skewness, and excess kurtosis of the lognormal dis-

tribution can be expressed in terms of the parameters s and m

mLN Â¼ exp (m Ã¾ 0:5s2 ),

s2 Â¼ [ exp (s2 ) Ã€ 1] exp (2m Ã¾ s2 ),

LN

SLN Â¼ [ exp (s2 ) Ã€ 1]1=2 [ exp (s2 ) Ã¾ 2],

KeLN Â¼ exp (4s2 ) Ã¾ 2 exp (3s2 ) Ã¾ 3 exp (2s2 ) Ã€ 6 (3:2:16)

The Cauchy distribution (Lorentzian) is an example of the stable distri-

bution (see the next section). It has the form

b

PC (x) Â¼ (3:2:17)

p[b2 Ã¾ (x Ã€ m)2 ]

The specific of the Cauchy distribution is that all its moments are

infinite. The case with b Â¼ 1 and m Â¼ 0 is named the standard Cauchy

distribution

1

PC (x) Â¼ (3:2:18)

p[1 Ã¾ x2 ]

Figure 3.1 depicts the distribution of the weekly returns of the ex-

change-traded fund SPDR that replicates the S&P 500 index (ticker

SPY) for 1996â€“2003 in comparison with standard normal distribution

and the standard Cauchy distributions (see Exercise 3).

The extreme value distributions can be introduced with the Fisher-

Tippett theorem. According to this theorem, if the cumulative distri-

bution function F(x) Â¼ Pr(X x) for a random variable X exists,

then the cumulative distribution of the maximum values of

X, Hj (x) Â¼ Pr(Xmax x) has the following asymptotic form

(

exp [Ã€(1 Ã¾ j(x Ã€ mmax )=smax )Ã€1=j ], j 6Â¼ 0,

Hj (x) Â¼ (3:2:19)

exp [Ã€ exp (Ã€(x Ã€ mmax )=smax )], jÂ¼0

where 1 Ã¾ j(x Ã€ mmax )=smax > 0 in the case with j 6Â¼ 0: In (3.2.19),

mmax and smax are the location and scale parameters, respectively;

j is the shape parameter and 1=j is named the tail index. The

24 Probability Distributions

0.5

0.4

0.3

0.2

SPY

Normal

Cauchy

0.1

0

âˆ’6 âˆ’4 âˆ’2 0 2 4

Figure 3.1 The standardized distribution of the weekly returns of the S&P

500 SPDR (SPY) for 1996â€“2003 in comparison with the standard normal

and the standard Cauchy distributions.

Fisher-Tippett theorem does not define the values of the parameters

mmax and smax . However, special methods have been developed for

their estimation [5].

It is said that the cumulative distribution function F(x) is in the

domain of attraction of Hj (x). The tail behavior of the distribution

F(x) defines the shape parameter. The Gumbel distribution corres-

ponds to the case with j Â¼ 0. Distributions with thin tails, such as

normal, lognormal, and exponential distributions, have the Gumbel

domain of attraction. The case with j > 0 is named the Frechet

distribution. Domain of the Frechet attraction corresponds to distri-

butions with fat tails, such as the Cauchy distribution and the Pareto

distribution (see the next Section). Finally, the case with j < 0 defines

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