Lecture_19

# Lecture_19 - 1 CHI-SQUARE DISTRIBUTION Lecture 19...

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1 CHI-SQUARE DISTRIBUTION Lecture 19 ORIE3500/5500 Summer2009 Chen Class Today Relatives of the Normal Distribution Transformation of Random Variables (CDF) Relatives of the normal distribution 1 Chi-square distribution If Y 1 ,...,Y n are i.i.d. standard normal variables, then the random variable X = Y 2 1 + ··· + Y 2 n is called a chi-square random variable with n degrees of freedom. If X has chi-squared distribution we write it as X χ 2 n . We can compute the mean and variance of the chi-square distribution with n degrees of freedom using the characterization from normal distribution: EY 2 1 = 1 and EY 4 1 = 3, E ( X ) = E ( Y 2 1 + ··· + Y 2 n ) = n. var ( X ) = var ( Y 2 1 ) + ··· + var ( Y 2 n ) = 2 n. It can be shown that the chi-square distribution with 1 degree of freedom is the same as Gamma(1/2,1/2) distribution, that is, it has density f X ( x ) = 1 2Γ(1 / 2) x - 1 / 2 e - x/ 2 ,x > 0 . We know that the sum of independent Gamma random variables with same scale parameter is also a Gamma random variable . Hence χ 2 n distrib- ution is same as the

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Lecture_19 - 1 CHI-SQUARE DISTRIBUTION Lecture 19...

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