kinds of distribution

kinds of distribution (statistics)

1. BY UNSA SHAKIR
DIFFERENT KINDS OF
DISTIBUTIONS

2. Bayes’ Theorem
Independent
Dependent

3. Ifdangerousfiresarerare(1%)butsmokeis
fairlycommon(10%)duetobarbecues,and
90%ofdangerousfiresmakesmokethen:
Example
Find "Probability of dangerous Fire when there
is Smoke“ using bayes theorem

4. P(Fire|Smoke) =P(Fire) P(Smoke|Fire)
P(Smoke)
=1% x 90%
10%
= 9%

5. Anentomologistspotswhatmightbea
raresubspeciesofbeetle,duetothepatternonitsback.
Intheraresubspecies,98%havethepattern,
orP(Pattern|Rare)=98%.
Inthecommonsubspecies,5%havethepattern.
orP(Pattern|common)=5%.
Theraresubspeciesaccountsforonly0.1%ofthe
population.P(Rare)=0.1%.P(common)=0.9%.
Howlikelyisthebeetlehavingthepatterntoberare,or
whatisP(Rare|Pattern)?
Example

7. Random variables
Arandomvariableissomenumerical
outcomesofarandomprocess
Arandomvariableisavariablewhose
valuesaredeterminedbytheoutcomeofa
randomexperiment.arandomvariableis
alsocalledachancevariable,astochastic
variableorsimplyavariate.

8. Random Variable
Examples of random variables:
•Let X be a random variable defined as sum of
dots when two dice are rolled. X can assumes the
values:2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
•LetYbearandomvariable,numberofheads
whenthreecoinsaretossed,thenthevaluesofY
willbe0,1,2,3

9. Probability Distribution:
Ifallpossiblevaluesofarandomvariable
canbewrittenalongwiththeirassociated
probabilities,thenthedistributioniscalled
probabilitydistribution.

10. Example
Roll a die,
Below showing the possible outcomes of
each number
x 1 2 3 4 5 6
f(x) 1/6 1/6 1/6 1/6 1/6 1/6

11. Example
Toss a coin twice. X= # of heads
xP(x)
0P(TT)=P(T)*P(T)=1/2*1/2=1/4
1P(TH or HT)=P(TH)+P(HT)=1/2*1/2+1/2*1/2=1/2
2P (HH)=P(H)*P(H)=1/2*1/2=1/4

12. Example:
LetXbearandomvariabledefinedas“Numberof
headswhentwocoinsaretossed”.Constructthe
probabilitydistributionofX.
Solution:Sample space when two coins are tossed:
S={HH,HT,TH,TT} XP(X)
01/4.
12/4.
21/4.
1

13. Q.LetXbearandomvariabledefinedassumofdots
whentwodicearerolled.Constructtheprobability
distributionofX
x p(x)
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36

14. Let X be a random variable defined as the difference
between dots when two dice are rolled, then
# X: 0, 1, 2, 3, 4, 5
(6,1)
(2, 4) (3, 4) (4, 4) (5,4)
(2,5)(3, 5)(4,5)(5,5)
(2, 6) (3, 6)(4,6)(5,6)
(1,1)(2,1)(3,1)(4,1) 

(1, 2) (2, 2) (3, 2)(4, 2) (6, 2)


(1,3)(2,3)(3,3)(4,3)
(5,1)
(5,2)

(5,3)(6, 3)
S
(1, 4) (6, 4)

 
(1,5) (6,5)
 
(1,6) (6,6)
X
0
1
2
3
4
5
P(X)
6/36
10/36
8/36
6/36
4/36
2/36

15. Verifythatforthenumberofheadsobtainedin
fourflipsofabalancedcointheprobability
distribution
Whatistheprobabilityofobtaining2heads
fromacointhatwastossed4times?
Examples

17. Types of RandomVariable
•Discrete RandomVariable
•Continuous RandomVariable
Types of ProbabilityDistributions
•Discrete ProbabilityDistributions
•Continuous ProbabilityDistributions

18. •Discrete ProbabilityDistributions
–Binomial ProbabilityDistribution
–Poisson ProbabilityDistribution
•Continuous ProbabilityDistributions
–Normal ProbabilityDistribution
–Exponential ProbabilityDistribution

19. Binomial Probabilitydistribution
Anexperimentissaidtobebinomialifitpossesses
thefollowingfourproperties
•Thereareonlytwopossibleoutcomes“Success”
and“Failure”
•Theprobabilityofsuccessdenotedbypremains
constantthroughoutthetrials
•Numberoftrialsarefixed
•Trialsareindependent

20. •P(Success)=p.
•P(Failure)=q=1-p.
•nindicatesthefixednumberoftrials.
•pindicatestheprobabilityofsuccessfor
anyonetrial.
•qindicatestheprobabilityoffailure(not
success)foranyonetrial.

21. What is P(x) for binomial?xnx
qp
xnx
n
xP



)!(!
!
)(
Where
p: probability ofsuccess
q: probability offailure
n: number oftrials
x: number ofsuccesses

22. ??????(�) = ??????
??????
�
�
1 − �
??????−�
= ??????
??????
�
�
�
??????−�
�ℎ??????�??????�= 0, 1, 2,⋯, ??????

23. Afaircoinistossed7times.
Findtheprobabilitiesofobtainingvarious
numbersofheads.
•n=7
•p=probability of appearinghead=1/2
•q=1-p=1-1/2=1/2
•X = 0,1,2,3,4,5,6,7
Example:

24. Solution:

30. Normal Approximation
Forlargen,thebinomialdistributioncan
beapproximatedbythenormal,is
approximatelystandardnormalforlarge
n.npq
npX
Z



31. Example
SupposeXisabinomialrandomvariable
withn=100andp=0.3.EstimateP(X=40)
byusingthenormalapproximation.

32. Poisson distribution
Eventshappenindependentlyintimeor
spacewith,onaverage,λeventsperunittime
orspace.
Radioactivedecay
λ=2particlesperminute
Lighteningstrikes
λ=0.01strikesperacre

33. Poisson probabilities
Under perfectly random occurrences it can be
shown that mathematically
Eventhappenindependentlyintimeorspacewith,
onaverage,λeventsperunitorspace( ) , x=0, 1, 2, ...
!
x
e
fx
x





34. Example
Radioactive decay
x= 3 # of particles/min
λ=2 particles per minutes32
2
( 3) , x=0, 1, 2, ...
3!
e
Px



35. Example
Radioactive decay
X=# of particles/hour
λ=2 particles/min * 60min/hour=120 particles/hr125 120
120
( 125) , x=0, 1, 2, ...
125!
e
Px



36. Exercise
Amailroomclerkissupposedtosend6of15
packagestoEuropebyairmail,buthegets
themallmixedupandrandomlyputsairmail
postageon6ofthepackages.Whatisthe
probabilitythatonlythreeofthepackages
thataresupposedtogobyairgetairmail
postage?

37. Exercise
Amonganambulanceservice’s16
ambulances,fiveemitexcessiveamountsof
pollutants.Ifeightoftheambulancesare
randomlypickedforinspection,whatisthe
probabilitythatthissamplewillincludeat
leastthreeoftheambulancesthatemit
excessiveamountsofpollutants?

38. Exercise
Thenumberofmonthlybreakdownsofthekindof
computerusedbyanofficeisarandomvariable
havingthePoissondistributionwithλ=1.6.Findthe
probabilitiesthatthiskindofcomputerwill
functionforamonth
Withoutabreakdown;
Withonebreakdown;
Withtwobreakdowns.