D ist r ibu si Var iab el A cak K ont inyu
5.5 D ist r ibu si Var iab el A cak K ont inyu
Cont ineous random variables can be used t o describe phenomena where t he values of a random variable can be any value in an int erval: t he t ime t o failure, or t he lengt h of a broken rod. Seven dist ribut ions will be discussed.
5.5.1 D i st r i busi U ni for m
a·x·b
f (x) =
b¡ a
0 ot her wi se 0 ot her wi se
² t he int erval where X can assume value can be arbit rarily long, but it cannot be in… nit e.
² Not e t hat x 2 ¡x 1
P (x 1 <X<x 2 ) = F (x 2 ) ¡ F (x 1 )=
b¡a is proport ional t o t he lengt h of t he int erval, for allx 1 dan x 2 st at isfying
a·x 1 <x 2 · b. ² Example 6.15 and 6.16 on page 202, 203
5.5.2 D i st r i busi Eksp onesi al
Exponent ial dist ribut ed random variable is oneof most frequent ly used dist ri- but ion in comput er simulat ion. I t is widely used in simulat ions of comput er network and ot hers.
0 ot her wise
² cdf
1¡e ¡¸x x¸0
F (x) =
0 x<0
² mean
E (X ) = 1=¸
² memoryless propert y of t he exponent ial dist ribut ed random variables: t he fut ure values of t he exponent ially dist ribut ed values arenot a¤ect ed by t he past values. Compare t his t o, for example, a uniformly dist rib- ut ed r andom variable, one can see t he di¤erence. For example, when t hrowing a fair coin, we can consider t he pr obabilit y of head and t ail is t he same which has t he value of 0.5. If, aft er a result of head, we would expect t o see a t ail (t hough it may not happen). I n exponent ially dis- t ribut ed random variable, we cannot have t his type of expect at ion. I n anot her word, we know not hing about t he fut ure value of t he random variable given a full hist ory of t he past .
Mat hemat ical proof. P (X > s + t ) e ¡ ¸ (s+ t)
² Example6.17 and 6.18 on page204 and 205 where Example 6.18 demon-
st rat es t he memoryless propert y of t he exponent ial dist ribut ion.
5.5.3 D i st r i busi Gam m a
0 other wise where ¡ (¯ ) = (¯ ¡ 1)! when ¯ is an int eger. ² When ¯ = = 1, t his is t he exponent ial dist ribut ion. I n anot her word,
t he Gamma dist ribut ion is a more general form of exponent ial dist rib- ut ion.
² mean
E (X ) = 1=µ
² variance
V (X ) = 2 ¯µ
5.5.4 D i st r i busi Erl ang
² When t he paramet er ¯ in Gamma dist ribut ion is an int eger, t he dist ri- but ion is refered t o as Erlang dist ribut ion.
² When ¯ = k, a posit ive int eger, t he cdf of Erlang dist ribut ion is (using int egrat ion by part s)
k¡ 1 P e ¡ kµx (kµx)!
0 x·0 which is t he sum of Poisson t erms wit h mean ² mean
² Example 6.19, 6.20 on page 208, 209.
5.5.5 D i st r i busi N or m al
This value is very di¢ cult t o calculat e. Oft en a t able is made for
X » N (0; 1). Because an X » N (¹ ; ¾ 2 ) can be t ransformed int o
X » N (0; 1) by let
X¡¹ Z= ¾
µ ¶ x¡¹ ² To calculat e P(X · x) for X » N (¹ ; ¾ 2 ), we use © ¾
Example: t o calculat e F (56) for N (50; 9), we have
² mean ¹ ² variance ¾ 2
² Not at ion: X » N (¹ ; ¾ 2 )
² The curve shape of t he normal pdf is like a ” bell” . ² propert ies:
– f (¹ ¡ x) = f (¹ + x) t he pdf is symmet ric about ¹ because of t his, ©(¡ x) = 1 ¡ ©(x)
– t he maximum value of t he pdf occurs at x = ¹ (t hus, t he mean and t he mode are equal.
² Example 6.21 and 6.22 on page 211, 6.23 and 6.24 on page 213.
5.5.6 D i st r i busi W ei bul l
The random variable X has a Weibull dist ribut ion if it s pdf has t he form
0 ot her wise ² Weibull dist ribut ion has t he following t hree paramet ers:
– ° which has t he range of which is t he locat ion paramet er – ® which is great er t han zero which is t he scale paramet er
– ¯ which is a posit ive value det ermines t he shape