Types of Statistical Distributions

7/26/2019 Types of Statistical Distributions

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COMMON STATISTICALDISTRIBUTIONS

Summary by: Gernimo Maldonado-Martnez Biostatistician

Data Management & Statistical Research Support nitni!ersidad "entral del "aribe


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Remember hypothesistesting?

#nly a smallprobability $% '() o*

getting a result

this small

#nly a smallprobability $% '() o*

getting a result

this largeResult could +easily, ha!e ariseni* there as no real di.erence

bet een groups

/

z


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What happens if the istrib!tionof i"eren#es #hanges a $itt$e?

Result could +easily, ha!e ariseni* there as no di.erence

bet een groups

0 much largerprobability o*

getting a result

this high1

0 much largerprobability o*

getting a result

this small/


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What is a istrib!tion?

2he complete summary o* the *re3uencieso* the !alues or categories o* ameasurement made on a group o* sub4ects

2he distribution sho s either ho many orhat proportion o* the group as *ound to

ha!e each !alue5 or a range o* !alues5 outo* all possible !alues

2he pattern o* !ariation o* a !ariable iscalled its distribution5 hich can bedescribed both mathematically andgraphically

6ast 7 M A dictionary of epidemiology #8*ord ni!ersity


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Types of %ariab$e !se here

"ontinuous ;rom to < =8: >eight5 ?gB count

Discrete ;inite number =8: @ o* heads & tails in a coin Aip


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Types of Distrib!tions

Binomial9oissonGamma

ormal=8ponential

t-distribution;-distribution"hi-s3uareddistribution?yper geometric6aplace


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Binomia$ Distrib!tion0 random se3uence o* n $C8ed) Bernoulli trials

;or each indi!idual trial#nly % possible outcomes $yes no5 heads tails)#utcome o* each trial is independent

9robability o* each outcome does not change o!er time9robability Mass ;unction $ x E number o*successes) the most *re3uently encountered in statistics ;or a C8ed number o* trials and each trial results in a

+success, ith probability p and a +*ailure, ith probability-p

xn x p p x

n x p = )1()(


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Shape of Binomia$ Distrib!tion

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10

x

p ( x )

00.05

0.10.15

0.20.25

0.30.35

0.4

0 5 10 15 20 25 30 35 40 45 50

x

p ( x )

0

0.05

0.1

0.150.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10

x

p ( x )

00.05

0.10.15

0.20.25

0.30.35

0.4

0 5 10 15 20 25 30 35 40 45 50

x

p ( x )

n =10 p =.15

n =10 p =.5

n =50 p =.15

n =50 p =.5

Shapes epen s great$y on si&e of n


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'oisson Distrib!tionFmportant and idely used

sed to model the number o* randomoccurrences o* an e!ent in a

continuous inter!al o* time or space=8amples: 9atients arri!ing =R umber o* a gi!en accident "ounts o* li!e or dead organisms 9article emissions *rom radioacti!e

source "alls arri!ing at a s itchboard


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'oisson Distrib!tion6et H E the a!erage number o*times that a repeated e!entoccurs per !nit of time or

spa#e under inspectionH determines the shape o* the9oisson distribution

=8ample: =mergencies "entroMIdicoH E JK per day or H E L per eeN


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'oisson Distrib!tion

9robability Mass;unction $8 Enumber o* e!ents)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

x

p ( x )

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

x

p ( x )

=1.97 =13.8

= e

x x p

x

!)(


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Re$ationship bet(eenBinomia$ an 'oisson

Distrib!tion>hen n is large and p is small5 a9oisson distribution can be usedto appro8imate a Binomialdistribution by letting = np=8ampleSetting up a ne burns unit *or allincidents in!ol!ing children 2o helpdecide on resource allocation eneed to Nno the !arious e8pectedprobabilities o* number o* patientsadmitted to the unit per day


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)amma Distrib!tionOery comple8 and !aried shapes9ro!ides a *airly Ae8ible class *ormodeling#ther Nno n distributions $eg=8ponential) are special cases of theGamma distribution #ther important distributions that are

special cases o* a gamma distribution andused regularly include chi-s3uared

Density ;unction P depends on %parameters


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)amma Distrib!tion

0

0.5

1

1.5

0 1 2 3 4 5 6

x

f ( x )

=4 =1

=2 =1

=1 =1

=0.5 =1

Shape o* !arious GammaDistributions


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Contin!o!s Distrib!tions

Statistical distributions that may taNeon a continuous range o* !alues?a!e a mathematical e3uation called aDensity ;unction5 f(x) *or an outcomef(x) must satis*ySometimes called "ontinuous9robability ;unction

=

=

1)(

realallfor0)(

)(][

dx x f

x x f

dx x f b xa P a

b


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What oes this mean?Density *unctions are deCned *oran inCnite number o* points o!era continuous inter!al

2he area under the cur!ebet een % distinct points deCnesthe probability that an outcome

*alls in that inter!al9robabilities are measured o!erinter!als and not single points


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Dis#rete Distrib!tions0 statistical distribution that can only takefnite or countable number of values"an deCne a mathematical e3uation

called a 9robability Mass ;unction5 p(x) p(x) must satis*y:

the prob that 8 can that a speciCc !alue is p$8)

1)(

xrealallfor0)(][)(

=

==

ii

i

ii

x p

x p x X P x p


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*+amp$e of Density,!n#tion (x)

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

*$8)

Ft is no only sensible to talN about theprobability o* an obser!ation *alling in an

inter!al


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'robabi$ity Mass ,!n#tion

0 coin is tossed L times0ll possible outcomes are ???5 ??25?225 ?2?5 22?5 2?25 2?? and 222F* 8 E number o* heads a*ter the Ltosses then9$8E/) E

9$8E ) E L9$8E%) E L9$8EL) E

/ % L


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Berno!$$i Ran om -ariab$e

#utcome taNe on only % !alues ithprobability p and 1-p

xample - !es " #o$ %eads " &ails

9robability Mass ;unction

10if ,0)(

1)0()1(

or x x p

p p p p

=

==


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*+ponentia$ Distrib!tion

"an be used to model aitingtimes or li*etimesShape depends on a singleparameter HQ/

H E mean aiting time per unit o*time

=8amples>aiting time =RSur!i!al time o* cancer patients

>orNing li*etime o* machine


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*+ponentia$ Distrib!tion

Density ;unction

0

0.5

1

1.5

2

0 1 2 3 4

x

f ( x

)=0.5

=1

=2

It has a mean of ./0 an a %arian#e of ./ 0 1

Types of Statistical Distributions

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