CHAPTER 2 – DISCRETE DISTRIBUTIONS HÜSEYIN GÜLER MATHEMATICAL STATISTICS Discrete Distributions...

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CHAPTER 2 – DISCRETE DISTRIBUTIONS HÜSEYIN GÜLER MATHEMATICAL STATISTICS Discrete Distributions 1

Transcript of CHAPTER 2 – DISCRETE DISTRIBUTIONS HÜSEYIN GÜLER MATHEMATICAL STATISTICS Discrete Distributions...

Page 1: CHAPTER 2 – DISCRETE DISTRIBUTIONS HÜSEYIN GÜLER MATHEMATICAL STATISTICS Discrete Distributions 1.

CHAPTER 2 – DISCRETE DISTRIBUTIONSHÜSEYIN GÜLER

MATHEMATICAL STATISTICS

Discrete Distributions

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Page 2: CHAPTER 2 – DISCRETE DISTRIBUTIONS HÜSEYIN GÜLER MATHEMATICAL STATISTICS Discrete Distributions 1.

2.1. DISCRETE PROBABILITY DISTRIBUTIONS

• The concept of random variable:

• S: Space or support of an experiment

• A random variable (r.v.) X is a real valued function defined on the space.

• X: S → R

• x: Represents the value of X

• x ε S

• X is a discrete r.v. if its possible values are finite, or countably infinite.

Discrete Distributions

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• A chip is selected randomly from the bowl:

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S = {1, 2, 3, 4} X: The number on the selected chip X is a r.v. with space S x = 1, 2, 3, 4. X is a discrete r.v. (it takes 4

different values)

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• P(X = x): Represents the probability that X is equal to x.

Discrete Distributions

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104

4 ,103

3 ,102

2 ,101

1 XPXPXPXP

The distribution of probability on the support S

xXPxf

4,3,2,1 ,10

xx

xfThe probability mass

function (p.m.f.)

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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

The probability histogram of X

1 2 3 4

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CALCULATING PROBABILITIES USING P.M.F.

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AxAx

xfxXPAP

Compute the probability that the number on the chip is 3 or 4.

10/710/410/34343 ffXP

4,3,2,1 ,10

xx

xf

If A is a subset of S then

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CALCULATING PROBABILITIES USING P.M.F.

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Compute the probability that the number on the chip is less than or equal to 3.

10/610/310/210/1

3213

fffXP

4,3,2,1 ,10

xx

xf

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RELATIVE FREQUENCIES AND RELATIVE FREQUENCY

HISTOGRAM

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n

xXxh

that timesof number the

The histogram of relative frequencies is called relative frequency histogram.

Relative frequencies converge to the p.m.f as n increases.

When the experiment is performed n times the relative frequency of x is

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• The chip experiment is repeated n = 1000 times using a computer simulation.

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x 1 2 3 4 Total

Frequency

98 209 305 388 n = 1000

h(x) 0.098 0.209 0.305 0.388 1

0

0.1

0.2

0.3

0.4

The relative frequency histogram of X

1 2 3 4

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THE COMPARISON OF f(x) AND h(x)

f(x) h(x)

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4Discrete Distributions

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f(x) is theoretically obtained while h(x) is obtained from a sample.

x 1 2 3 4 Total

f(x) 0.1 0.2 0.3 0.4 1

h(x) 0.098 0.209 0.305 0.388 1

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THE MEAN OF THE (PROBABILITY) DISTRIBUTION

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34.043.032.021.01

4

1

x

xxf

called the mean of X.

It is possible to estimate μ using relative frequencies.

The weighted average of X is

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THE MEAN OF THE EMPIRICAL DISTRIBUTION

• x1, x2,..., xn: Observed values of x

• fj: The frequency of uj

• uj = 1, 2, 3, 4.

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983.2388.04305.03209.02098.01

11 4

1

4

11

xj

jj

n

ii xxhuf

nx

nx

xh the empirical distribution the mean of the empirical distribution or the

sample meanx

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THE VARIANCE AND THE STANDARD DEVIATION OF THE

DISTRIBUTION

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1

4034303320321031 2222

22

....

x

xfxXVar

The variance of X is

112

The standart deviation of X is

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AN ALTERNATIVE FOR THE VARIANCE OF THE DISTRIBUTION

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1

3404303202101 22222

22

22

....

x

x

xfx

xfxXVar

x

r xfx r_th moment about the origin

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THE VARIANCE OF THE EMPIRICAL DISTRIBUTION

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9890

983238804305032090209801 22222

22

2

......

xxhx

xhXx

x

x

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THE VARIANCE AND THE STANDART DEVIATION OF THE

SAMPLE

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99009890999

10001

2 ..

nn

s

2

995099002 .. ss

s2 (the variance of the sample) is an estimate of (the variance of X).