chapter 2 – discrete distributions hÜseyin gÜler mathematical statistics discrete distributions...
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CHAPTER 2 – DISCRETE DISTRIBUTIONSHÜSEYIN 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
2
• A chip is selected randomly from the bowl:
Discrete Distributions
3
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)
• P(X = x): Represents the probability that X is equal to x.
Discrete Distributions
4
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.)
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
Discrete Distributions
5
CALCULATING PROBABILITIES USING P.M.F.
Discrete Distributions
6
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
CALCULATING PROBABILITIES USING P.M.F.
Discrete Distributions
7
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
RELATIVE FREQUENCIES AND RELATIVE FREQUENCY
HISTOGRAM
Discrete Distributions
8
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
• The chip experiment is repeated n = 1000 times using a computer simulation.
Discrete Distributions
9
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
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
10
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
THE MEAN OF THE (PROBABILITY) DISTRIBUTION
Discrete Distributions
11
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
THE MEAN OF THE EMPIRICAL DISTRIBUTION
• x1, x2,..., xn: Observed values of x
• fj: The frequency of uj
• uj = 1, 2, 3, 4.
Discrete Distributions
12
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
THE VARIANCE AND THE STANDARD DEVIATION OF THE
DISTRIBUTION
Discrete Distributions
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1
4034303320321031 2222
22
....
x
xfxXVar
The variance of X is
112
The standart deviation of X is
AN ALTERNATIVE FOR THE VARIANCE OF THE DISTRIBUTION
Discrete Distributions
14
1
3404303202101 22222
22
22
....
x
x
xfx
xfxXVar
x
r xfx r_th moment about the origin
THE VARIANCE OF THE EMPIRICAL DISTRIBUTION
Discrete Distributions
15
9890
983238804305032090209801 22222
22
2
......
xxhx
xhXx
x
x
THE VARIANCE AND THE STANDART DEVIATION OF THE
SAMPLE
Discrete Distributions
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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).
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