Normal distribution

1 / 9

# Normal distribution - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Normal distribution' - ivana

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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, 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 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

f(x)

μ

σ

x

Normal distribution

X  N(μ, σ2)

Extension belowthecurveisthe total probability (=1).

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

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 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).