probability density function concept figure 7.8 consider a signal that varies in time. what is the...
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Probability Density Function Concept
Figure 7.8
• Consider a signal that varies in time.
• What is the probability that the signal at a future time will reside between x and x + x?
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0 2 4 6 8 10 12 140
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
x
p(x)
Probability Density Function
In-Class Example
130
13713
711
10
)(361
361
x
xx
xx
x
xp
5.0
1177
1
)(]71Pr[
2212
21
361
71
221
361
7
1361
7
1
xx
dxx
dxxpx
• Determine the probability that x is between 1 and 7.
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Probability Distribution Function (PDF)• The probability distribution function (PDF) is related to the integral of the pdf.
• A consequence of this is that
Figure 7.14
pdf PDF
corrections
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Normalization of the pdf
1)( dxxpWhen the pdf is ‘normalized’ correctly:
Here,
3)( dxxp
>> not normalized
So, define pnew(x) = 1/3 p(x)
such that
1)( dxxpnew
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Probability Density & Distribution
pdf PDF
Fig. 7.14
Pr[x≤1] =
0.5 / 3.0 =1/6 = 16.7 %
• Determine Pr[x≤1] using the pdf and using the PDF.
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In-Class Example• Determine the expressions for the PDF curve, knowing that of the pdf curve.
13713
71)(
721332
21
361
7212
21
361
xxx
xxxxP
• Integrating the pdf expressions give
0 2 4 6 8 10 12 140
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
x
p(x)
Probability Density Function
130
13713
711
10
)(361
361
x
xx
xx
x
xp
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
P(x
)
Probability Distribution Function
5.0
05.0
)1()7(
)(]71Pr[7
1
PP
dxxpx
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In-Class Example
N2 molecules travel at speeds in this room according to the Maxwell-Boltzman speed distribution. The probability density functionthat describes this is:
Determine the speed below which 95 % of the molecules travel.
Tk
mCC
Tk
mCp
BB 2exp
24)(
22
2/3
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In-Class Example
∫→
pdf PDF
810 m/s
C =
p
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The Normal pdf and PDF
Figure 8.4