Proofs related to chi | chi square distribution pdf
Thefollowingareproofsofseveralcharacteristicsrelatedtothechi-squareddistribution.Derivationsofthepdf[edit]Derivationofthepdfforonedegreeoffreedom[edit]LetrandomvariableYbedefinedasY=X2whereXhasnormaldistributionwithmean0andvariance1(thatisX ~ N(0,1)).Then,for y<0, FY(y)=P(Y
The following are proofs of several characteristics related to the chi-squared distribution.
Derivations of the pdf[edit] Derivation of the pdf for one degree of freedom[edit]Let random variable Y be defined as Y = X2 where X has normal distribution with mean 0 and variance 1 (that is X ~ N(0,1)).
Then,for y<0, FY(y)=P(Y<y)=0 andfor y≥0, FY(y)=P(Y<y)=P(X2<y)=P(|X|<y)=P(−y<X<y) =FX(y)−FX(−y)=FX(y)−(1−FX(y))=2FX(y)−1{displaystyle {egin{alignedat}{2}{ ext{for}}~y<0,&~~F_{Y}(y)=P(Y<y)=0~~{ ext{and}}\{ ext{for}}~ygeq 0,&~~F_{Y}(y)=P(Y<y)=P(X{2}<y)=P(|X|<{sqrt {y}})=P(-{sqrt {y}}<X<{sqrt {y}})\~~&=F_{X}({sqrt {y}})-F_{X}(-{sqrt {y}})=F_{X}({sqrt {y}})-(1-F_{X}({sqrt {y}}))=2F_{X}({sqrt {y}})-1end{alignedat}}}
fY(y)=ddyFY(y)=2ddyFX(y)−0=2ddy(∫−∞y12πe−t22dt)=212πe−y2(y)y′=212πe−y2(12y−12)=1212Γ(12)y−12e−y2{displaystyle {egin{aligned}f_{Y}(y)&={ frac {d}{dy}}F_{Y}(y)=2{ frac {d}{dy}}F_{...