Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT...
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Transcript of Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT...
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