Chi square test for mathematical statistics, Study notes of Mathematical Statistics

Download Chi square test for mathematical statistics and more Study notes Mathematical Statistics in PDF only on Docsity!

Exact Sampling Distributions (Chi-square Distribution) 13-1. Chi-Square Variate (Pronounced as Ki - Sky without S). The square of a staridard normal variate is known as a chi-square variate with 1 degree of freedom (d.£) Thus if X ~ W (1, 0%, then Z =~ ~ No, 1) ad Za (4) , is achi-square variate with 1 d.f. In general, if X;, = 1, 2; ..., 4) are independent normal variates with mean pi; and variance G7, (i= 1,2,...,), then gs “shy. isa chi-square variate witha d.f. —...(13-1) iwi 13-2, Derivation of the Chi-square Distribution. First Method—Methed of Moment Generating Function. 1fX;, @= 1,2, ..., m) are independent N(u;, 67), we want the distribution of =a z uh) = = 5, U2, where Uys ist Since X;’s are independent, U;’s are also independent. MO=Msy,O= T tl ty: = My, OF, since U;'’s ~ N (O, 1) are identically dissibuted Now Myg(d =Bloxp (UN) =]” expt’) dex =| pur) wn” (- Gi Wo?) dy; “-— J £x9 uP) x0 uP) ds [x 222]