4. Probability Distribution

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Random Variables and

Probability Distribution

Purnomo

Jurusan Teknik MesinUGM


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Random VariablesA random variable X is a numerical valued

function defined on a sample space.

A number X(e), providing a measure ofcharacteristic of interest, is assigned toeach simple event e in the sample

spaceContoh dadu :X = 1, 2, 3, 4, 5, 6


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Two balls are drawn in succession from a box that contains 4red balls and 3 blue balls. The possible outcomes and the valuesy of the random variable Y, where Y is the number of red balls

is

Sample space y

RR 2

RB 1

BR 1

BB 0


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IllustrationTwo products A and B are judge by four consumer who then

expressed a preference for A and B. The outcome when the firstand third consumers prefer A and the other consumers prefer B

is denoted by ABAB. The number of outcomes is 24 = 16.

AAAA AAAB AABB ABBB BBBB

AABA ABAB BABB

ABAA ABBA BBAB

BAAA BAAB BBBA

BBAA

BABA


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IllustrationSuppose that the products are alike in quality and that the

consumers express their preference independently. Then the 16simple events in the sample space are equally likely, and each

has a probability of 1/16. Let a random variable X be devined asX= number of person preffering A to B.Probability distribution :

Distinct value of X 0 1 2 3 4

Probability 1/16 4/16 6/16 4/16 1/16

P[X2] = 6/16 + 4/16 + 1/16 = 11/16P[1X3] = 4/16 + 6/16 + 4/16 = 14/16


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Probability Distribution

The probability distribution or simply, the distribution ofa discrete random variable is a list of the distinctvalues of xi together with their associate probabilitiesf(xi) = P[X=xi]


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Graphic PresentationLine diagram

x

f(x)

0 1 2 3 4

2/16

4/16

6/16


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Histogram of probability Histogram

Value x 1 2 3 4

f(x) 1/8 1/8

1 2 3 4 5

2/8

4/8Area = 0.5


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Properties of relative frequency histogram

The total area under the histogram is 1

For the two points a and b such that each is a

boundary point of some class, the relativefrequency of measurements in the interval ato b is the area under the histogram enclosedby this interval.


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ExpectationExpected value or Expectation of X

)()( ii xfxXE !

X 0 1 2 3 4 5 Totalf(x) 0.1 0.1 0.2 0.3 0.2 0.1 1

xf(X) 0 0.1 0.4 0.9 0.8 0.5 2.7

E(X) = 2.7

E(X) = population mean =


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Variance : a measure of spreadDeviation = X

(x1 ), (x2 ), .(xk )

Probabilitiesf(x1), f(x2), .f(xk)

E(deviation) = E(X ) = (xi )f(xi) = 0

Deviation can not be used as a measure of spread


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Variance and standard

Deviation

Variance of X (= 2 = x2)

Var(X) = E[(X )2] = E(X2) 2

Standard deviation ( = = x )

sd(X) = Var (X)


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Standardized random variable

Standardized random variable :

Random variable Z has a mean of 0 andvariance of 1

k

kX

ZW

Q! has E(Z) = 0 and Var(Z) = 1

Bentuk ini akan banyak digunakan pada applikasi


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PROBABILITY MODELS FOR CONTINUOUSRANDOM VARIABLES

The probability distribution of acontinuous random variable can bevisualized as a smooth form of relativefrequency histogram based on largenumber of observations.


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PROBABILITY DENSITY CURVE

Probability density curve can be viewedas a limiting form of relative frequencyhistogram(number of classes - infinite )

P2


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Slide 15

P2 PURNOMO, 8/23/2006


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Properties of Probability Density Function, f(x)

The total area under the density curve is 1

area under thedensity curve between a and b

f(x) is positive or zero

For continuous random variable, the

probability that X=x is always 0 (X is onlymeaningful when X lies in an interval

? A!ee bXaP


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Density Curves Measuring center and spread for density

curves

Density curves describe the overall shape of adistribution

Ideal patterns that are accurate enough forpractical purposes

Faster to draw and easier to use

Areas or proportions under the curverepresent counts or percents of observations


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Features of a Continuous Distribution

As with relative frequency histograms, the probabilitydensity curves of continuous random variables posses

a wide variety of shapes :- Negatively skewed

- Symmetric

- Positively skewed

- Flat- Bell shaped

- Peaked


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Center of a Density Curve The mode of a distribution is the point where

the curve is highest

The median is the point where half of thearea under the curve lies on the left and theother half on the right. Equal Areas Point

Quartiles can be found by dividing the area

under the curve into four equal parts of the area is to the left of the 1stquartile of the area is to the left of the 3rd quartile

The mean is the balance point.4

3


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Percentiles

Percentiles are defined as :

The population 100p-th percentile is an x value that has

an area p to the left and 1-p to the right.

Lower (first) quartile = 25th percentile

Second quartile (or median) = 50th percentile

Upper (third) quartile=75th percentile


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The Normal distribution Discovered by Abraham de Moivre around 1720. Around 1870,

Adolph Quetelet realised that the normal curve could be used tocompare histograms of data.

Chest measurements of 5738 Scottish soldiers by Belgianscholar Lambert Quetelet (1796-1874)

Pierre Laplace dan Carl Gauss : bell-shaped distribution

Gauss derived the normal distribution mathematically as theprobability distribution of the error of measurements, which is

called normal law of error Gaussian Distribution


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Normal Distributions Symmetric Single-peaked (unimodal) Bell-shaped The mean, median, and mode are the same The points where there is a change in

curvature is one standard deviation on eitherside of the mean.

The mean and standard deviation completelyspecify the curve


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Normal Distribution

The height of a normaldensity curve at any pointxis given by

2)(

2

1

2

1)( W

Q

TW

!x

exf

is the mean

is the standard deviation

Q

W

Q

W

),( WQN


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The Empirical Rule 68% of the observations fall within one

standard deviation of the mean

95% of the observations fall within twostandard deviation of the mean

99.7% of the observations fall withinthree standard deviation of the mean


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Example: Young Womens

Height The heights of young women are approximately

normal with mean = 64.5 inches and std.dev. = 2.5

inches.


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The normal distribution is the most important distribution

in Statistics. Typical normal curves with different sigma

(standard deviation) values are shown below.


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Examples with approximate

Normal distributions Height

Weight

IQ scores

Standardized test scores

Body temperature Repeated measurement of same

quantity


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FACTS Universality of the normal distribution is

only a myth, and examples of quitenonnormal distribution abound in anyvirtually every field of study

Still, the normal distribution plays a

central role in statistics (make thingseasier)


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Standardizing and z-Scores One case, one curve --- too

complicated

Solution -- standardization

normalization

non-dimensionalization

---- z-Scores

---- All cases, one curve (or table)


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Standardizing and z-Scores an observation x comes from a distribution with

mean and standard deviation The standardized value ofx is defined as

which is also called az-

sco

re. A z-score indicates how many standard deviations

the original observation is away from the mean,and in which direction.

,

W

!

xz


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The Standard Normal Curve

N(0,1)


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The Standard Normal Table The Normal Table is a table of areas under the

standard normal density curve. The table entry for eachvalue z is the area under the curve to the left ofz.


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The Standard Normal Table The Normal Table can be used to find the proportion of

observations of a variable which fall to the left of a specificvalue z if the variable follows a normal distribution.


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Use of The Normal TableArea under curve to the left of z

(area to the left of b)- (area to theleft of a)

? A !e zZP

? A!ee bzaP


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Use of The Normal Table If z>0

? A ? A

? A ? AzZPzZP

zZPzZP

e!e

e!e

05.0

05.0


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Use of The Normal Table Calculate z

Find the area to the left of z in StandardNormal Probability Table

Other calculations obey the propertiesof the Standardized Normal Curve


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Example : random variable


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Example : Expectation


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4. Probability Distribution