Simple%20stochastic%20models%202

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Simple stochastic models 2
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Continuous random variable X

• X can take values -? lt x lt ?
• Cumulative probability distribution function
• PX(x) P(X ? x)
• Probability density function

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Normal distribution
X normal(?,?), E(X) ?, V(X) ? 2

• pdf

cpdf
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?0, ?1
0.4
0.3
pdf
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
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cpdf
x
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Central limit theorem

• Y X1 X2 Xn
• Xi - independent, zero mean, equal variance V
• If n is large then

normal(0,1)
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Example students body heights
Histogram of students body heights versus normal
probability density function
160
165
170
175
180
185
190
195
200
205
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Example children birth weights
Histogram of children birth weights versus normal
probability density function
-4
x 10
0
1000
2000
3000
4000
5000
6000
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Binomial becomes normal
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
30
35
40
45
50
o binomial(0.5,50)
normal(25,3.5355)
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How do we fit normal distribution to data ?

• Data X1, X2, , Xn

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• How do we estimate parameters of distributions
  using data ?
• How do we verify that data follow a given
  distribution ?

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Characteristic function

• X with pdf p(x)
• characteristic function

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Properties
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Properties
YaXb, a,b - constants
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Characteristic function of normal distribution

• X normal(?,?),

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Continuity theorem for characteristic functions
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Two dimensional distributions

• X, Y
• Probability density function p(x,y)

Cumulative pdf
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Independent random variables

• X, Y independent
• pXY(x,y)pX(x) pY(y)
• Convolution integral ZXY

pZ pX pY
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Convolution and characteristic functions
ZXY
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Use of characteristic functions to prove Central
Limit Theorem

• Y X1 X2 Xn

i1,2,n
so
and
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