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...
![Page 1: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/1.jpg)
Continuous Continuous DistributionDistribution
![Page 2: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/2.jpg)
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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)
![Page 4: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/4.jpg)
![Page 5: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/5.jpg)
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
![Page 6: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/6.jpg)
2. Gamma Distr2. Gamma Distr
2
1
!1)(
1),1(1)(
Theorema
nn
![Page 7: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/7.jpg)
PDF GAMMA FUNCTIONPDF GAMMA FUNCTION
0,)(
1,;,,~ 1
xexxfGAMXx
2
XV
XE
![Page 8: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/8.jpg)
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
![Page 9: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/9.jpg)
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
![Page 10: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/10.jpg)
A Rv X with pdf :
2
2
2
,Notation
σ,
,0 and
,,parameter with
,2
1),;(
2
2
N
XVXE
xexfx
![Page 11: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/11.jpg)
Standar Normal PDFStandar Normal PDF
)1,0(~
,2
1 then If 2
2
NZ
zeΦ(z)x
zz
)()1()(''
)()('
)()(
2 zzz
zzz
zz
![Page 12: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/12.jpg)
propertiesproperties
(z) has unique maximum at z=0Inflection points at z=1
![Page 13: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/13.jpg)
exampleexample
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Other exOther ex
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ThTh
If then
),(~ 2NX
xxF
NX
Z
X )(.2
1,0~.1
![Page 17: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/17.jpg)
exampleexample
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Other exOther ex
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Normal Approximation to the Binomial Normal Approximation to the Binomial DistributionDistribution
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exex
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Normal approximation to the Normal approximation to the Poisson DistributionPoisson Distribution
![Page 24: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/24.jpg)
exex
solutionsolution
![Page 25: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/25.jpg)
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
![Page 26: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/26.jpg)
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exex
solutionsolution
![Page 28: Continuous Distribution. 1. Continuous Uniform Distribution f(x) 1/(b-a) abx Mean : MIDPOINT Variance : square length I(a,b) A continuous rV X with.](https://reader033.fdocuments.us/reader033/viewer/2022042821/56649cae5503460f94971064/html5/thumbnails/28.jpg)
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Other distributionOther distributionPossibility new thema about other distribution
excluded in MATHSTAT :Erlang distributionLog normalChi KuadratGumbellTweedie RayleightBetaPearsonCauchyBenford