TSP distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ TSP(a, b, m, n) is used to indicate that the random variable X has the TSP distribution with parameters a, b, m, and n. A TSP random variable X has probability density function

f(x) = \{ \begin{matrix}\frac{n(x−a)^{n−1}}{(b−a) (m−a)^{n−1}} & a < x < m \\ \frac{n(b−x)^{n−1}}{(b−a) (b−m)^{n−1}} & m ≤ x < b, \end{matrix}

for n > 0, and a < m < b. The probability density function with a = 0,b = 8,m = 3, and n = 4 is illustrated below.

PSfrag replacements

x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \{ \begin{matrix}\frac{(x−a)^{n}(m−a)^{1−n}}{b−a} & a < x < m \\ −\frac{a+b(b−x)^{n}(b−m)^{−n} −b−m(b−x)^{n}(b−m)^{−n}}{b−a} & m ≤ x < b. \end{matrix}

The survivor function on the support of X is

S(x) = P(X ≥ x) = \{ \begin{matrix}−\frac{−b+a+(x−a)^{n}(m−a)^{1−n}}{b−a} & a < x < m \\ \frac{(b−x)^{n}(b−m)^{1−n}}{b−a} & m ≤ x < b. \end{matrix}

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \{ \begin{matrix}−\frac{n(x−a)^{n−1} (m−a)^{1−n}}{a−b+(m−a)(x−a)^{n}(m−a)^{−n}} & a < x < m \\ \frac{n}{b−x} & m ≤ x < b. \end{matrix}

The moment generating and characteristic functions of X are mathematically intractable. The population mean and variance of X are

E[X] = \frac{b − m + mn + a}{n + 1}

V [X] = \frac{−2mnb + 2m^{2}n + a^{2}n + b^{2}n − 2mna + 2bm − 2ba + 2am − 2m^{2}}{(n + 1) ^{2}(n + 2)} .

APPL verification: The APPL statements

X := [[x -¿ (n * (x - a) ˆ (n - 1)) / ((b - a) * (m - a) ˆ (n - 1)), x -¿ (n * (b - x) ˆ (n - 1)) / ((b - a) * (b - m) ˆ (n - 1))], [a, m, b],[”Continuous”, ”PDF”]];

CDF(X) SF(X); HF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

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