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?

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.

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

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

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.

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

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

In-Class Example

∫→

pdf PDF

810 m/s

C =

p

The Normal pdf and PDF

Figure 8.4