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Normal distribution

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

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  1. Normal distribution

  2. 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?

  3. 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

  4. 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

  5. f(x) μ σ x Normal distribution X  N(μ, σ2) Extension belowthecurveisthe total probability (=1).

  6. 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) =

  7. Normal distribution • Distribution function(u) ofthestandardizednormal distributionN(0, 1), μ= 0, σ=1

  8. 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).

  9. 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).

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