follows a t -distribution with r degrees of freedom. If Z N ( 0, 1) and U 2 ( r) are independent, then the random variable: T Z U / r. Lets just jump right in and define it Definition. x2/2 x2 > 0 is the pdf of the chi-square distribution with degrees of freedom n1 +n2. Properties: The density function of U is: fU (u) u 1/2 e u/2, 0 < u < 2 Recall the density of a Gamma(, ) distribution: g(x) () x 1 e x, x > 0,So U is Gamma(, ) with 1/2 and 1/2.Starting with your last questions first: yes, the $\chi^2$ distribution with $k$ degrees of freedom is normally defiined as being the sum of the squares of $k$ independent $N(0,1)$ distributions. We have just one more topic to tackle in this lesson, namely, Students t distribution. has a Chi-Squared distribution with 1 degree of freedom.
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