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Student's z-Distribution


StudentsZDistribution

The probability density function for Student's z-distribution is given by

 f_n(z)=(Gamma(n/2))/(sqrt(pi)Gamma((n-1)/2))(1+z^2)^(-n/2).
(1)

Now define

 d_n(z)=(|z|^(1-n)Gamma(1/2n)_2F_1(1/2(n-1),1/2n;1/2(n+1);-z^(-2)))/(2sqrt(pi)Gamma[1/2(n+1)]),
(2)

then the cumulative distribution function is given by

 D_n(z)={d_n(z)   for z<=0; 1-d_n(z)   for z>=0
(3)

The mean is 0, so the moments are

mu_1=0
(4)
mu_2=1/(n-3)
(5)
mu_3=0
(6)
mu_4=3/((n-3)(n-5)).
(7)

The mean, variance, skewness, and kurtosis excess are

mu=0
(8)
sigma^2=1/(n-3)
(9)
gamma_1=0
(10)
gamma_2=6/(n-5).
(11)

The characteristic function is

 phi(t)=(2^((3-n)/2)|t|^((n-1)/2)K_((1-n)/2)(|t|))/(Gamma[1/2(n-1)]),
(12)

where K_n(z) is a modified Bessel function of the second kind.

Letting

 z=(x^_-mu)/s,
(13)

where x is the sample mean and mu is the population mean gives Student's t-distribution.


See also

Student's t-Distribution

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References

Kenney, J. F. and Keeping, E. S. "'Student's' z-Distribution." §7.11 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 174-175, 1951.

Referenced on Wolfram|Alpha

Student's z-Distribution

Cite this as:

Weisstein, Eric W. "Student's z-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Studentsz-Distribution.html

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