Random Variable2013

Random VariableTwo types

Discrete

Continuous

Random VariableProbability mass functionDiscrete

P(X = xi) = p(xi)

p(xi) = 1

Random VariableProbability density functionContinuousf(x) = e –x x > 0P(X = a) = 0-

f(x) dx = 1

P(a < x < b) = ab f(x) dx

Random VariableExpected value

= E(x) = xi p (xi) = x f(x) dx

Random VariableVariance

))((

))((22

22

22

2

)(

]2[

])[()(

xpxx

xEx

iixip

E

xxE

xExV

i

Random VariableStandard deviation

Sums of R.V.)()( XVXSD

)()()(

)()()(

22

212

1

2211

2211

xVaxVaYV

xEaxEayE

xaxaY

Random Variable

n

XXSampleMean

i

11

)(222

2

n

n

n

XXanceSampleVari xx

S ii

Poisson Probability Distribution Consider a discrete r.v. which is often useful when dealing with the number of occurrences of an event over a specified interval of time. Suppose we want to find the probability distribution of the accidents at the intersection of Rural and Apache during a one week period. The R.V. we are interested in is the number of accidents.

Poisson Probability Distribution i. The Poisson Distribution provides a good model for the probability

distribution of the number of rare events that occur in space, time, and volume where is the average at which events occur.

ii. Define: A r.v. is said to have a Poisson distribution if the p.m.f of X is

P(x) = f(x) = !x

ex

, x = 0,1, …

where is the rate per unit time or per unit area

iii.

)(

][

XV

XE

Exponential Distribution Previously, we discussed the Poisson random variable, which was the number of events occurring in a given interval. This number was a discrete r.v. and the probabilities associated with it could be described by the Poisson Probability Distribution. Not only is the number of events a r.v., but the waiting time between event is also a random variable. This r.v. is a continuous r.v. for it can assume any positive value. This r.v. is an exponential r.v. which can be described by the exponential distribution.

Exponential Distribution i . P d f:

otherwise

xexf

x

0

0&0 )(

w h e re = ra te a t w h ic h e v e n ts o c c u r

i i . C o rre sp o n d in g ly ,

2

0

1)(

1][

0,1)()(

XV

XE

xedxexXPxF xx

x

i i i . A n im p o rta n t a p p lic a tio n o f th e e x p o n e n tia l d is tr ib u tio n is to

m o d e l th e d is tr ib u tio n o f c o m p o n e n t l ife tim e . A re a so n fo r i ts p o p u la rity is b e c a u se o f th e “ m e m o ry-le ss” p ro p e rty o f th e E x p o n e n tia l D is tr ib u tio n

The Uniform Distribution o The simplest distribution is the one in which a continuous r.v. can assume

any value within a interval [a, b]

Def: A continuous r.v. X is said to have a uniform distribution on the interval [a,b] if the probability distribution (pdf) of X is:

otherwise

bxaabxf

0

1)(

The Uniform Distribution The cumulative distribution is

12

)()(

2)

1()(][

)(

)()()(

2abXV

abdx

abxdxxxfXE

ab

ax

ab

a

ab

x

a

x

ab

xdxxf

dxxfxXPXF

xx

x

x

The Uniform Distribution Note: An important uniform distribution is that for when a = 0 and b = 1, namely U(0, 1) A U(0,1) r.v. can be used to simulate observation of other random variables of the discrete and continuous type.

The Triangular Distribution • Continuous Distribution

elsewhere

cxbacbc

xc

bxaacab

axxf

0

))((

)(2

))((

)(2)(

The Triangular Distribution

cx

cxbacbc

xc

bxaacab

axxF

axxF

1

))((

)(1

))((

)()(

0)(

2

2

The Triangular Distribution

18)(

3)(

0)(

222 bcacabcbaxV

cbaxE

axxF

Normal Distribution I t i s a f a c t t h a t m e a s u r e m e n t s o n m a n y r a n d o m v a r i a b l e s w i l l f o l l o w a b e l l -s h a p e d d i s t r i b u t i o n . R a n d o m v a r i a b l e o f t h i s t y p e a r e c l o s e l y a p p r o x i m a t e d b y a N o r m a l P r o b a b i l i t y D i s t r i b u t i o n . A c o n t i n u o u s r . v . X i s s a i d t o h a v e a n o r m a l d i s t r i b u t i o n i f t h e p d f o f X i s

,,0,2

1)(

2

2

2

)(

xexfx

T h e d i s t r i b u t i o n c o n t a i n s 2 p a r a m e t e r s ( a n d ) . T h e s e a r e t h e e x p e c t e d v a l u e a n d t h e v a r i a n c e a n d h e n c e l o c a t e t h e c e n t e r o f t h e d i s t r i b u t i o n a n d m e a s u r e i t s s p r e a d .

Normal Distribution T h e S t a n d a r d N o r m a l D i s t r i b u t i o n T o c o m p u t e P ( a x b ) w h e n X ~ N ( , 2 ) , w e m u s t e v a l u a t e

dxedxxfb

a

xb

a

2

2

2

)(

2

1)(

N o t e : N o n e o f t h e s t a n d a r d i n t e g r a t i o n t e c h n i q u e s c a n b e u s e d t o e v a l u a t e t h i s p d f . I n s t e a d , f o r = 0 , a n d 2 = 1 , t h e p d f h a s b e e n e v a l u a t e d a n d v a l u e s h a v e b e e n c o m p u t e d . U s i n g t h e t a b l e , p r o b a b i l i t i e s f o r a n y o t h e r v a l u e s o f a n d 2 c a n b e d e t e r m i n e d

Normal Distribution T h e n o r m a l d i s t r i b u t i o n f o r p a r a m e t e r s v a l u e s = 0 , a n d 2 = 1 i s c a l l e d t h e s t a n d a r d n o r m a l d i s t r i b u t i o n . A r . v . t h a t h a s a s t a n d a r d d i s t r i b u t i o n i s c a l l e d a s t a n d a r d n o r m a l r a n d o m v a r i a b l e ( d e n o t e d b y Z ) . T h e p d f o f Z i s :

zezfz

,2

1)( 2

2

Normal Distribution T h e c u m u l a t i v e d i s t r i b u t i o n o f Z i s

(Z)by denoted is and)()(

z

dyyfzZP

N o t e : T h e N ( 0 , 1 ) T a b l e r e t u r n s t h e c u m u l a t i v e p r o b a b i l i t y u p t o z o r ( z )

Selecting a Distribution Theoretical prior knowledge

Random arrival => exponential IATSum of large manufactures => Normal

CLTCompare histogram with probability mass

or probability density