Continuous Continuous DistributionDistribution

1. Continuous Uniform Distribution1. Continuous Uniform Distribution

otherwise0

1),;( bxa

abbaxf

f(x)

1/(b-a)

a b x

Mean : MIDPOINT

Variance :

square length I(a,b)

2

ba

12

22 ab

A continuous rV X with pdf

Is a continuous uniform rV or X~UNIF(a,b)

exampleexample

Let X Continuous rV denote the current measured in a thin cooper wire in mA. Assume that the range of X is [0,20mA] and assume that the pdf of X is

What is the probability that a measurement of current is between 5 and 10 mA?

200,05.0)( xxf

33.3312

2010

025.0)05.0(5)(105

2

10

5

XVXE

dxxfXP

2. Gamma Distr2. Gamma Distr

2

1

!1)(

1),1(1)(

Theorema

nn

PDF GAMMA FUNCTIONPDF GAMMA FUNCTION

0,)(

1,;,,~ 1

xexxfGAMXx

2

XV

XE

3. Normal Distribution (Gaussian)3. Normal Distribution (Gaussian)The most used model for the distribution of a rV ?

why?Whenever a random experiment is replicated, the

rV that equal the average (total) result over the replicates tends to have a normal distribution as the number of replicates become large De Moivre The CLT

Normal curve shapeNormal curve shape

n(x)

0

0.05

0.1

0.15

0.2

0.25

0.3

-6 -4 -2 0 2 4 6

x

μ

σ

A Rv X with pdf :

2

2

2

,Notation

σ,

,0 and

,,parameter with

,2

1),;(

2

2

N

XVXE

xexfx

Standar Normal PDFStandar Normal PDF

)1,0(~

,2

1 then If 2

2

NZ

zeΦ(z)x

zz

)()1()(''

)()('

)()(

2 zzz

zzz

zz

propertiesproperties

(z) has unique maximum at z=0Inflection points at z=1

exampleexample

Other exOther ex

ThTh

If then

),(~ 2NX

xxF

NX

Z

X )(.2

1,0~.1

exampleexample

Other exOther ex

Normal Approximation to the Binomial Normal Approximation to the Binomial DistributionDistribution

exex

Normal approximation to the Normal approximation to the Poisson DistributionPoisson Distribution

exex

solutionsolution

4. Exponential Distribution4. Exponential DistributionLet the rV N denote the number of flaws in

x mm of wire. If the mean number of flaws is per-mm, N has a Poisson distribution with mean x.

Now

0,)(

and

0,1)(

therefore!0

)(0

0

xexf

xexXPxF

exe

NPxXP

x

x

xx

exex

solutionsolution

Other distributionOther distributionPossibility new thema about other distribution

excluded in MATHSTAT :Erlang distributionLog normalChi KuadratGumbellTweedie RayleightBetaPearsonCauchyBenford