Arctangent distribution X - Mathematics
[Pages:2]Arctangent distribution (from ) The shorthand X arctan(, ) is used to indicate that the random variable X has the arctangent distribution with phase shift parameter and positive location parameter . An arctangent random variable X with parameters and has probability density function
f (x) =
(arctan ( ) + 1/2 ) 1 + 2 (x - )2
x0
for > 0 and - < < . The probability density function with three different choices of parameters is illustrated below.
f (x)
1.6
1.4
= 2, = 3
1.2
1.0
0.8 = 1, = 1
0.6
0.4 0.2
0.0
-4 -2 0 2
= 1, = 3
x
468
The cumulative distribution function of X is
F(x) = P(X x) = 2
arctan ( ) - arctan (-x + ) 2 arctan ( ) +
x 0.
The survivor function of X is
S(x)
=
P(X
x)
=
+ 2 arctan (-x + ) 2 arctan ( ) +
x 0.
The hazard function of X is
h(x)
=
f (x) S(x)
=
(1 + 2x2
2 - 2 2 x + 22) ( + 2
arctan (-x + ))
x 0.
The cumulative hazard function of X is
H(x) = ln (2 arctan ( ) + ) - ln ( + 2 arctan ( (-x + )))
x 0.
The inverse distribution function of X is
F
-1(u)
=
+
tan
(-
arctan
(
)
+
u
arctan (
)
+
1/2
u )
0 < u < 1.
1
The moments of X are undefined. It follows that the population mean, variance, skewness, and kurtosis of X are also undefined. APPL verification: The APPL statements X := ArcTanRV(lambda, phi); CDF(X); SF(X); HF(X); IDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X); verify the cumulative distribution function, survivor function, hazard function, population mean, variance, skewness, kurtosis, and moment generating function.
2
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