![]() ![]() The is always greater than or equal to zero i.e. The values of have been tabulated for different degrees of freedom at different levels of probability. If this number is large (>30), which generically happens for large samples, then the t-Student distribution is practically indistinguishable from N(0,1). The student distribution is well defined for any, but in practice, only positive integer values of are of interest. This distribution is similar to N(0,1), but its tails are fatter the exact shape depends on the number of degrees of freedom. Random variable has the student distribution with degrees of freedom. When n is small, the distribution is markedly different from normal distribution but as n increases the shape of the curve becomes more and more symmetrical and for n > 30, it can be approximated by a normal distribution. Suppose that has the standard normal distribution, has the chi-squared distribution with degrees of freedom, and that and are independent. The shape of distribution depends on the degrees of freedom which is also its mean (Fig.1). 2 converges to the Chi-square distribution with (k 1) degrees of freedom. Alternatively if a sample of size n, is drawn from a normal population with variance σ 2, the quantity (n-1) s 2 / σ 2 follows distribution with (n-1) degrees of freedom where s 2 is the sample variance. Let a random vector (n1 .,n k) have a Multinomial distribution with ni. If X 1, X 2………….X n are n independent standard normal variates, then sum of squares of these variates X 1 2+X 2 2 +…………………………+X n 2 follows the distribution with n degrees of freedom. Theoretically, Chi-square ( ) distribution can be defined as the sum of squares of independent normal variates. 7.1.Introduction to Chi-square ( ) distribution ![]()
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