2.4 The Normal, Chi-Squared, Student t, and F Distributions

Peeravich PAOPRAYOONHitotsubashi UniversityIntroduction to EconometricsNormal Distribution

Normal DistributionLots of things turn out to be approximately normal. (See the Central Limit Theorem)Most frequently used PDF in econometrics. R mean 2 > 0 variancePropertiesSingle peakedSymmetric about the mean; skewness is zeroKurtosis = 3Can be standardized to have mean 0 and variance 1

Normal Distribution

Any normal random variable Y N(,2) may be transformed to the standard normal random variable Z N(0,1) by subtracting the mean and dividing by the standard deviation. Z = (Y ) / Therefore, you only need one set of tables!Standard Normal DistributionJoint distribution2 variables: bivariate normal distributionPropertiesIf X and Y have a bivariate normal distribution with covariance XY and if and b are two constants

Marginal distribution is normalCovariance = 0 then variables are independentConditional expectation:

Multivariate Normal Distribution

Bivariate Normal Distribution

Used when testing certain types of hypothesesSum of m squared independent standard normal r.v.m is called degree of freedomChi-Squared Distribution

Chi-Squared Distributionm: degree of freedomRatio of a standard normal random variable divided by the square root of an independently distributed chi-squared random variable with m degrees of freedom divided by mStudent t Distribution

W: chi-squared with m V: chi-squared with nF Distribution

James H. Stock and Mark W. Watson, Introduction to Econometrics, 3rdhttps://econ.duke.edu/uploads/assets/Slides2.pdfWikipedia

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