Appendix A Probability Density Functions

APPENDIX A

Probability Density Functions

Frequently in the text, a random variable is said to have a certain type of probability density function, such as Gaussian, Cauchy, Laplace, uniform, or others. Since the exact form of the probability density function indicated by these labels is not perfectly standardized and since the reader may be unfamiliar with some of these probability densities, those probability density functions which appear often in the text are listed here for easy reference. Obviously, no attempt has been made at completeness, but rather the goal is to minimize confusion and thus to allow efficient utilization of the textual presentation.

Gaussian or Normal (Fig. A-1)

Parameters: I -

p, u

fX (X

1

P-sCr PQU P-@ P

210

.

x

W WUP+50

Flg. A-1. Example of a Gaussian density function.

ProbabiIity Density Functions

2zz

Uniform or Rectangular (Fig. A-2)

1,

fX(x)

Parameters:

a, b

1fx fl

1 b-a '

for a < x < b otherwise

- -----

'

b-a

,x

Fig. A-2. Example of a density function for a uniform distribution.

Cauchy (Fig. A-3)

Parameters:

A, 0

Fig. A-3.

Example of a Cauchy density function.

Laplace or Double Exponential (Fig. A-4) fX

Parameter:

( x ) = f h e-+l

- 00

< x < 00

A

Exponential (Fig. A-5) fx(x) =

Parameter:

for

A

("7,

for x > o otherwise

Appendix A

212

-2

0

-I

I

Fig. A-4. Example of a Laplace density function.

0

I

I

I

2

Ax

Fig. A-5. Example of a density function for an exponential distribution.

Binomial (Fig. A-6) f x ( x ) = P { X = k } = b(k; n , p ) = (;)p"qn-"

Parameters: n, p Student's t (Fig. A-7)

Parameter: n

for k = 0, 1 , 2 , .

. . , n.

Probability Density Functions

213

Fig. A-7. Example of a density function for a Student's r distribution.