T distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ t(n) is used to indicate that the random variable X has the t distribution with positive, real-valued parameter n, known as the degrees of freedom. In most applications, n is a positive integer. A t random variable X with n degrees of freedom has probability density function

f(x) = \frac{Γ((n + 1)/2)(1 + x^{2}/n)^{(−(n+1)/2)}}{\sqrt{nπ}Γ(n/2)} − ∞ < x < ∞.

The t distribution arises in hypothesis tests concerning the comparison of (a) a sample mean to a standard, or (b) the difference between two means. The probability density function for n = 1 (a special case of the t distribution known as the Cauchy distribution) and n = 5 is illustrated below.

PSfrag replacements

n = 1
n = 5
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = ∫_{−∞}^{x} \frac{Γ((n + 1)/2)n^{n/2}(n + t^{2})^{((n−1)/2)}}{\sqrt{π}Γ(n/2)} dt − ∞ < x < ∞.

The survivor function on the support of X is

S(x) = P(X ≥ x) = ∫_{x}^{∞} \frac{Γ((n + 1)/2)n^{n/2}(n + t^{2})^{((n−1)/2)}}{\sqrt{π}Γ(n/2)} dt − ∞ < x < ∞.

The hazard function and cumulative hazard function on the support of X is mathematically intractable.

The inverse distribution function of X is mathematically intractable.

The median of X is 0 because f(x) is an even function.

The moment generating function of X is undefined, although various moments exist for restricted values of the parameter n.

The population mean, variance, skewness, and kurtosis of X are

E[X] = 0 for n > 1, V [X] = \frac{n}{n − 2} for n > 2,

E [( \frac{X − μ}{σ} ) ^{3}] = 0 for n > 3, E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3(n − 2)}{n − 4} for n > 4.

APPL verification: The APPL statements

X := TRV(n); CDF(X); SF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the cumulative distribution function, survivor function, hazard function, population mean, variance, skewness, and kurtosis.