D ist r ibu si Var iab el A cak D isk r i t

5.4 D ist r ibu si Var iab el A cak D isk r i t

Here we are going t o st udy a few discret e random variable dist ribut ions.

5.4.1 B er noul l i t r i al s dan di st r i busi B ernoul l i

² A Bernoulli t rail is an experiment wit h result of success or failure. ² We can use a random var iable t o model t his phenomena. Let X j =1

if it is a success, X j = 0 if it is a failure. ² A consecut ive n Bernoulli t rails are called a Bernoulli process if

– t he t r ails are independent of each ot her; – each t rail has only t wo possible out comes (success or failure, t rue

² The following relat ions hold. p(x 1 ;x 2 ;:::;x n )=p 1 (x 1 ) ¢p 2 (x 2 ) ¢: : : p n (x n )

which means t he probabilit y of t he result of a sequence of event s is equal t o t he product of t he probabilit ies of each event .

Because t he event s are independent and t he probabilit y remains t he same,

x j = 1; j = 1; 2; : : : ; n p j (x j ) = p(x j )= 1¡p=qx j = 0; j = 1; 2; : : : ; n

² Not e t hat t he ” locat ion” of t he p i ’s don’t mat t er. I t is t he count of p i ’s t hat is import ant .

² Examples of Bernoulli t rails include: a conscut ive t hrowing of a ” fair” coin, count ing heads and t ails; a pass or fail t est on a sequence of a part icular component s of t he ” same” qualit y; and ot hers of t he similar t ype.

² For one t rial, t he dist ribut ion above is called t he Bernoulli dist ribut ion. The mean and t he variance is as follows.

E(X j ) = 0 ¢q + 1 ¢p = p

V (X j )=E X ¡ (E [X ]) = 0 ¢q + 1 ¢p ¡p = p(1 ¡ p)

5.4.2 D i st r i busi B i nomi al

² The number of successes in n Bernoulli t rials is a random variable, X . ² What is t he probabilit y t hat m out of n t rials ar e success?

m p n¡ m

m;n =p q

² There are

n!

x!(n ¡ x)!

² So t he t ot al probability of m successes out of n t r ials is given by

p x q n¡ x x = 0; 1; 2; : : : ; n

² Mean and variance: consider t he binomial dist ribut ion as t he sum of n independent Bernoulli t rials. Thus

E [X ] = p + p + : : : + p = np

V (X ) = pq + pq + : : : + pq = npq ² Example 6.10 on page 198

5.4.3 D i st r i busi Geom et r i k

² The number of t rials needed in a Bernoulli t r ial t o achieve t he …rst success is a random variable t hat follows t he geomet ric dist ribut ion.

² The dist ribut ion is given by

½ q x¡1 p x = 1; 2; : : : p(x) =

0 ot her wi se

² The mean is calculat ed as follows.

because t he sequence converges, so we can exchange t he order of sum- mat ion and t he di¤erent iat ion.

² Example 6.11 on page 199

5.4.4 D i st r i busi Poi sson

The Poisson dist ribut ion has t he following pdf

8 < e ¡® ® x x = 0; 1; : : :

p(x) =

x!

0 other wise

wher e ® > 0 ² Poisson dist ribut ion propert y: mean and var iance are t he same

E(X ) = ® = V(X )

² The cdf of t he Poisson dist ribut ion

because ® is a const ant and t he increase rat e of i ! is much fast er t han t hat of ® i , t he dist ribut ion is most ly det ermined by t he … rst a few it ems.

² Poisson dist ribut ion is most useful in Poisson processwhich isused most

frequent ly in describing such phenomena as t elephone call arrivals. ² Example 6.12, 6.13, 6.14 on page 200, page 201