Student's z-Distribution

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StudentsZDistribution

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

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

(1)

Now define

[画像: 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

[画像: 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 =

(4)

mu_2 = [画像:1/(n-3)]

(5)

mu_3 =

(6)

mu_4 = [画像:3/((n-3)(n-5)).]

(7)

The mean, variance, skewness, and kurtosis excess are

mu =

(8)

sigma^2 = [画像:1/(n-3)]

(9)

gamma_1 =

(10)

gamma_2 = [画像:6/(n-5).]

(11)

The characteristic function is

[画像: 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

[画像: 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.

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Student's z-Distribution

Cite this as:

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

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