Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 128

n

þ n e k : Notice also that lim n !1 1 ¼ This implies that,

for X n Binomial ð n ; k= n Þ and X Poisson ðkÞ; lim n !1 pX n ¼ ½ k ¼ pX ½ ¼ k :

For X Poisson( ʻ ),

*

EX ðÞ ¼ k

and

V ðÞ ¼ k:

A.7.2 Examples of Continuous Probability Distributions

A.7.2.1 Uniform Distribution

$

X has a Uniform distribution on the interval [a, b] (X * Uniform(a,b)) X has the

density f X determined by
f X x

ðÞ ¼ ½ 1 = ð b a Þ 1 ½a ; b x ðÞ for a x b and 0 otherwise :

\ \

That means, the density of x depends only on x being inside or outside the
interval [a, b]. So, this density has a rectangular graph and

EX ¼ ð a þ b Þ= 2 :

A.7.2.2 Triangular Distribution

X has a triangular distribution on the interval [a, b] with mode M, for M ∈ [a, b]

(X Triangular(a,M,b)) $

*

f X x ðÞ ¼ ½ ð x a Þ= ð M a Þ 2 = ð b a Þ for a x M ;

\

f X x ðÞ ¼ ½ ð b x Þ= ð b M Þ 2 = ð b a Þ for M x b

\

and

f X x ðÞ ¼ 0 otherwise :

Triangular(a,M,b), the mode of X is M and if M = (a+b)/2, then X is
symmetric around M, that is with expected value, mode and median equal to M.
The expectation of X is

If X *

EX ¼ ð a þ b þ M Þ= 3 :