SOME SPECIFIC PROBABILITY DISTRIBUTIONS E(x-)2=(=+)h=0 (#a(=)+a2(f)k= The third central moment is E(x-)2=需( (t2 () (2°(=)+2to +3 g6 +3tσ 0 The fourth central moment i E(X-p)4=(-)h σ8(e)+3t2o +326(e (=) +6t2a +3a(e2 2. CHI-SQUARE RANDOM VARIABLE 2.1. Probability Density Function. The random variable X is said to be a chi-square random variable with v degrees of freedom (abbreviated x(u)] if the density function of X is given by )=2r( 20 (10) 0 otherwise here r(.is the gamma function defined by r(r)=fo ur 0 (11) Note that for positive integer values of r, r(r)=(r-1)!SOME SPECIFIC PROBABILITY DISTRIBUTIONS 5 E (X − µ )2 = d2 dt2  e t2 σ2 2  |t = 0 = d dt  t σ2  e t2 σ2 2  |t= 0 =  t2 σ4  e t2 σ2 2  + σ2  e t2 σ2 2  |t= 0 = σ2 (7) The third central moment is E (X − µ )3 = d3 dt3  e t2 σ2 2  |t= 0 = d dt  t2 σ4  e t2 σ2 2  + σ2  e t2 σ2 2  |t = 0 =  t 3 σ6  e t2 σ2 2  + 2 t σ4  e t2 σ2 2  + t σ4  e t2 σ2 2  |t= 0 =  t 3 σ6  e t2 σ2 2  + 3 t σ4  e t2 σ2 2  |t = 0 = 0 (8) The fourth central moment is E (X − µ )4 = d4 dt4  e t2 σ2 2  |t = 0 = d dt  t 3 σ6  e t2 σ2 2  + 3 t σ4  e t2 σ2 2  |t = 0 =  t 4 σ8  e t2 σ2 2  + 3 t 2 σ6  e t2 σ2 2  + 3 t 2 σ6  e t2 σ2 2  + 3 σ4  e t2 σ2 2  |t = 0 =  t4 σ8  e t2 σ2 2  + 6 t2 σ6  e t2 σ2 2  + 3 σ4  e t2 σ2 2  |t = 0 = 3 σ4 (9) 2. Chi-square random variable 2.1. Probability Density Function. The random variable X is said to be a chi-square random variable with ν degrees of freedom [abbreviated χ2(ν) ] if the density function of X is given by f (x ; ν) = 1 2 ν 2 Γ ( v 2 ) x ν−2 2 e −x 2 0 < x = 0 otherwise (10) where Γ ( · ) is the gamma function defined by Γ (r ) = R ∞ 0 u r − 1 e −u du r > 0 (11) Note that for positive integer values of r, Γ(r) = (r - 1)!