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Chapter 2 Normal Distribution The normal distribution is one of the most important continuous probabilitydistributions, and is widely used in statistics and other ﬁelds of sciences. In thischapter, we present some basic ideas, deﬁnitions, and properties of normal distrib-ution, (for details, see, for example, Whittaker and Robinson (1967), Feller (1968,1971), Patel et al. (1976), Patel and Read (1982), Johnson et al. (1994), Evans etal. (2000), Balakrishnan and Nevzorov (2003), and Kapadia et al. (2005), amongothers). 2.1 Normal Distribution The normal distribution describes a family of continuous probability distributions,having the same general shape, and differing in their location (that is, the mean oraverage)andscaleparameters(thatis,thestandarddeviation).Thegraphofitsproba-bilitydensityfunctionisasymmetricandbell-shapedcurve.Thedevelopment ofthegeneral theories of the normal distributions began with the work of de Moivre (1733,1738) in his studies of approximations to certain binomial distributions for largepositive integer  n  >  0. Further developments continued with the contributions of Legendre(1805),Gauss(1809),Laplace(1812),Bessel(1818,1838),Bravais(1846),Airy (1861), Galton (1875, 1889), Helmert (1876), Tchebyshev (1890), Edgeworth(1883, 1892, 1905), Pearson (1896), Markov (1899, 1900), Lyapunov (1901), Char-lier (1905), and Fisher (1930, 1931), among others. For further discussions on thehistory of the normal distribution and its development, readers are referred to Pear-son (1967), Patel and Read (1982), Johnson et al. (1994), and Stigler (1999), andreferences therein. Also, see Wiper et al. (2005), for recent developments. The nor-mal distribution plays a vital role in many applied problems of biology, economics,engineering, ﬁnancial risk management, genetics, hydrology, mechanics, medicine,numbertheory,statistics,physics,psychology,reliability,etc.,andhasbeenhasbeenextensively studied, both from theoretical and applications point of view, by manyresearchers, since its inception. M. Ahsanullah et al.,  Normal and Student’s t Distributions and Their Applications , 7Atlantis Studies in Probability and Statistics 4, DOI: 10.2991/978-94-6239-061-4_2,© Atlantis Press and the authors 2014  8 2 Normal Distribution  2.1.1 Deﬁnition (Normal Distribution) A continuous random variable  X   is said to have a normal distribution, with mean  µ andvariance σ  2 ,thatis,  X   ∼  N  (µ, σ  2 ) ,ifitspdf   f   X  (  x  ) andcdf   F   X  (  x  )  =  P (  X   ≤  x  ) are, respectively, given by  f   X  (  x  )  =  1 σ  √  2 π e − (  x  − µ) 2 / 2 σ  2 ,  −∞  <  x   <  ∞ ,  (2.1)and F   X  (  x  )  =  1 σ  √  2 π  x    −∞ e − (  y − µ) 2 / 2 σ  2 dy =  12[1  +  er f    x   −  µσ  √  2  ,  −∞  <  x   <  ∞ ,  −∞  < µ <  ∞ , σ >  0 ,  (2.2)where  erf  (.)  denotes error function, and  µ  and  σ  are location and scale parameters,respectively.  2.1.2 Deﬁnition (Standard Normal Distribution) A normal distribution with  µ  =  0 and  σ   =  1, that is,  X   ∼  N  ( 0 , 1 ) , is called thestandard normal distribution. The pdf   f   X  (  x  )  and cdf   F   X  (  x  )  of   X   ∼  N  ( 0 , 1 )  are,respectively, given by  f   X  (  x  )  =  1 √  2 π e −  x   2 / 2 ,  −∞  <  x   <  ∞ ,  (2.3)and F   X  (  x  )  =  1 √  2 π  x    −∞ e − t  2 / 2 dt  ,  −∞  <  x   <  ∞ , =  12  1 + erf    x  √  2  ,  −∞  <  x   <  ∞ .  (2.4)Note that if   Z   ∼  N  ( 0 ,  1 )  and  X   =  µ + σ   Z  , then  X   ∼  N  (µ,σ  2 ) , and converselyif   X   ∼  N  (µ, σ  2 )  and  Z   =  (  X   −  µ)/ σ  , then  Z   ∼  N  ( 0 ,  1 ) . Thus, the pdf of any general  X   ∼  N  (µ, σ  2 )  can easily be obtained from the pdf of   Z   ∼  N  ( 0 ,  1 ) ,by using the simple location and scale transformation, that is,  X   =  µ  +  σ   Z  . Todescribetheshapesofthenormaldistribution,theplotsofthepdf (2.1)and cdf (2.2),  2.1 Normal Distribution 9 Fig. 2.1  Plots of the normalpdf, for different values of   µ and  σ  2 for different values of   µ  and  σ  2 , are provided in Figs.2.1 and 2.2, respectively, by using Maple 10. The effects of the parameters,  µ  and  σ  2 , can easily be seen fromthese graphs. Similar plots can be drawn for other values of the parameters. It isclear from Fig.2.1 that the graph of the pdf   f   X  (  x  )  of a normal random variable,  X   ∼  N  (µ, σ  2 ) , is symmetric about mean,  µ , that is  f   X  (µ  +  x  )  =  f   X  (µ  −  x  ) , −∞  <  x   <  ∞ .  2.1.3 Some Properties of the Normal Distribution This section discusses the mode, moment generating function, cumulants, moments,mean,variance,coefﬁcientsofskewnessandkurtosis,andentropyofthenormaldis-tribution,  N  (µ, σ  2 ) . For detailed derivations of these, see, for example, Kendall andStuart (1958), Lukacs (1972), Dudewicz and Mishra (1988), Johnson et al. (1994),RohatgiandSaleh(2001),BalakrishnanandNevzorov(2003),Kapadiaetal.(2005),and Mukhopadhyay (2006), among others.  10 2 Normal Distribution Fig. 2.2  Plots of the normalcdf for different values of   µ and  σ  2 2.1.3.1 Mode The mode or modal value is that value of X for which the normal probability densityfunction  f   X   (  x  )  deﬁned by (2.1) is maximum. Now, differentiating with respect to xEq.(2.1), we have  f     X   (  x  )  = −   2 π  (  x   −  µ)  e −  (  x  − µ) 2 / 2 σ  2 σ  3  , which, when equated to 0, easily gives the mode to be  x   =  µ , which is the mean,that is, the location parameter of the normal distribution. It can be easily seen that  f     X   (  x  ) <  0. Consequently, the maximum value of the normal probability densityfunction  f   X   (  x  )  from (2.1) is easily obtained as  f   X   (µ)  =  1 σ  √  2  π . Since  f    (  x  )  =  0has one root, the normal probability density function (2.1) is unimodal. 2.1.3.2 Cumulants The cumulants  k  r   of a random variable X are deﬁned via the cumulant generatingfunction

Jul 23, 2017

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