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

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download