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Normal distribution. f. X. Normal distribution. Are equally distributed random variables common in reality?. How does the distribution of a variable observable in nature typically look like?. f(x ). Normal distribution N(μ, σ 2 ) . μ. x. μ , σ . x = μ . Normal distribution.

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Normal distribution

Normal distribution


Normal distribution1

f

X

Normal distribution

Are equally distributed random variables common in reality?

How does the distribution of a variable observable in nature typically look like?


Normal distribution2

f(x)

Normal distribution N(μ, σ2)

μ

x

μ , σ

x = μ 

Normal distribution

Parameters:

x    f(x)  0

x  -  f(x)  0

f(x) maximal for

f(x) symmetric around μ

In R: dnorm() Densityfunction

pnorm()Distribution function

qnorm() Quantiles


Normal distribution3
Normal distribution

  • Example: bloodpressure

X  N(μ=140, σ2=100)

X  N(μ=140, σ2=25)

X  N(μ=160, σ2=100)

X  N(μ=160, σ2=225)

Alteration of μ  translation on the x-axis

Alteration σ  dilationofthecurve


Normal distribution4

f(x)

μ

σ

x

Normal distribution

X  N(μ, σ2)

Extension belowthecurveisthe total probability (=1).


Normal distribution5

f

(

x

)

150

x

[mmHg]

120

140

160

180

Normal distribution

  • Question: Whatistheporbability, that a patienthas a bloodpressure <= 150 mmHg ?(whenbloodpressureisnormallydistributedwithμ=140 andσ2=100)

P (X ≤ 150) =


Normal distribution6
Normal distribution

  • Distribution function(u) ofthestandardizednormal distributionN(0, 1), μ= 0, σ=1


Normal distribution7

Standardized normal distribution

Normal distribution

N(μ, σ2)

N(0, 1)

Normal distribution

  • Each normal distribution N(μ, σ2) witharbitraryμandσ² canbetransformedintothestandardized normal distribution N(0, 1).


Normal distribution8
Normal distribution

  • Central limittheorem:

    LetX1,X2,…,Xnbeindependentandidenticallydistributedrandom variables withE(Xi)=μandVar(Xi)=σ² für i=1,..,n. Then, thedistributionfunctionsoftherandom variables sn=converge against thedistributionfunctionΦofthestandard normal distribution0,1).