Probability Distribution Function.ppt

8/14/2019 Probability Distribution Function.ppt

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Probability Distribution

In any Probabilistic situation each strategy(course ofaction) may lead to a number of different possibleoutcomes.

A listing of all the possible outcomes of a random variable with each outcome`s associated probabilityof occurrence is called Probability Distribution.


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ontd..

!he numerical value of a random variable dependsupon the outcome of an e"periment and may bedifferent for different trails of same e"periment .

!he set of all such values so obtained is called therange space of the random variable.

In all such cases# the decision ma$er may li$e to$now

%. !he avg. value of the random variable&. !he ris$ involved in choosing a strategy.


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'"ample

If a coin tossed twice# then the sample space ofevents# for this random e"periment is

* ++# !+# +!# !!,

If the decision ma$er is interested to $now theprobability distribution for the number of heads#then a random variable (") may be defines as-

" no. of +`s occurred


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!he values of " will depend on chance and may ta$ethe values- ++ +! %# !+ %# !! .

!hus the range space of " is * #%#&,.

/hen a random variable " is defined# a value is givento each simple event in sample space.

!he probability of any particular value of " can then be found by adding the probabilities for all thesimple events that have the value of ".


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uppose P(" r) represents the probability of random variable ta$ing the value r (here r represents the no.of heads occurred).


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0andom 1ariable

A random variable x ta$es on a defined set of values withdifferent probabilities.

2 For e"ample# if you roll a dice# the outcome is random (not fi"ed)

and there are 3 possible outcomes# each of which occur withprobability one4si"th.

2 For e"ample# if you poll people about their voting preferences# thepercentage of the sample that responds 5es on Proposition % is a also a random variable (the percentage will be slightly

differently every time you poll).


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0oughly# probability is how fre6uently we e"pect differentoutcomes to occur if we repeat the e"periment over andover ( fre6uentist view)


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0andom variables can be discrete or continuous

Discrete random variables have a countablenumber of outcomes

'"amples- Dead7alive# treatment# dice# counts# etc.

Contin o s random variables have an infinitecontinuum of possible values.

'"amples- blood pressure# weight# the speed of a car# etc.


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Prob!bi"it# $ nctions

A probability function maps the possible values of x againsttheir respective probabilities of occurrence# p(x)

p(x) is a number from to %. .

!he area under a probability function is always %.


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Probability Distribution Function(pdf)

It can be classified in two categories-

%. Discrete probability distributions&. ontinuous probability distributions


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Discrete Probability Distribution

It assumes that the outcomes of a random variableunder study can ta$e on only integer values

A boo$ shop has only #%#.. opies of a particulartitle of a boo$

A consumer can buy #%#X shirts# pants etc.


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If the random variable " is discrete# its probabilitydistribution called probability mass function 9 itmust satisfy the following condition-

!he probability of a any specific outcome for a discrete random variable must be between and %. : f(" $) : %# for all value of $

!he sum of the probabilities over all possible values of a discreterandom variable must e6ual %. ; f(" $) % all $


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Discrete e"ample- roll of a dice

x

p(x)

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1 4 5 62 3

= xall

1P(x)


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Probability mass function (pmf)

x p(x)1 p(x=1) =1/6

2 p(x=2) =1/6

3 p(x=3) =1/6

4 p(x=4) =1/6

5 p(x=5) =1/6

6 p(x=6) =1/61.0


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ontinuous Probability Distribution

It assumes that the outcomes of random variablecan ta$e on only value in an interval

Product costs and prices. Floor area of a house# office etc


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If the random variable " is continuous# then itsprobability density function must satisfy twoconditions-

P(") < = 4> ? " ? > (non4negative condition)

P(")d" % (Area under the continuous curvemust total %)


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!his function is used to find probabilities associated with random variable values @% # @.. @n in agiveninterval or range say (a#b).

In other words these probabilities are determined byfinding the area under the pdf between the values aand b.

f(a : " : b) f(b) f(a)


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'"ample

onsider the function a :":B f(")

otherwise

is this a pdf C


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olution

For f(") to be a pdf# the condition

f(") d" % must be satisfied# which is true

if B

a d" %#

i.e. a %7B.

ince a # the function# f(") < . !hus f(") satisfies both the conditions for pdf.


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Euestion

A given right angle triangle is a pdf. o find out theheight of the trangle.


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If this is a pdf# so the area must be e6ual to %

i.e. A bGh %

given# b H so# GH G h % h &7H Ans


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Probability Distribution Function.ppt

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