Ece3075 a 8

1. ECE 3075A
Random Signals
Lecture 8
Probability Distribution Functions, Probability
Density Functions
School of Electrical and Computer Engineering
Georgia Institute of Technology
Fall, 2003
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #1
Probability Density Functions
• The slope of the probability distribution function at x
represents the incremental probability at that point and
thus gives the sense of how likely X = x might be.
f x FX x FX x dFX x
X
( ) lim ( ) ( ) ( )
dx
ε ε
→ 0
ε
=
+ − +
=
ε
F (x +ε ) X
F (x) X
f (x)dx Pr(x X x dx) mass X = < ≤ +
ε
Probability
is the probability mass at x.
• The derivative is called the probability density function
(pdf). Pdf is non-negative. In the case of discrete
distributions, the pdf consists of Dirac delta functions
at those realizable values, each having an area equal
to the corresponding magnitude of probability.
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #2
Examples of PDFs
• Distribution of a discrete
r.v.
• Continuous r.v. –
uniform distribution
x
F (x) X
1
x
f (x) X
1/(b-a)
a b
F (x) X
1
0.23
0.86
0.5
f (x) X
0.23 0.27 0.36
0.14
-1 0 0.5 1.5
=Σ −
x
x
X i i f (x) pδ (x x ) f x b a a x b X ( ) = ( − )−1, ≤ ≤
i
∞
∫ f X ( x ) dx = F
X (∞) =1 −∞ Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #3
Gaussian Random Variable
• A r.v. X is gaussian if its pdf is of the form

2
f x = x X x X ,
∞ < < ∞ − 
− −
( ) 1 2
exp ( )

2
πσ σ
2
X and σ 2
where are called the mean and variance, respectively.
σ = standard deviation
f (x) X F (x) X
Peak
0.607max Max slope
X -σ X X +σ
x
max = ( 2πσ )−1
X -σ X X +σ
x
1
0.841
0.5
0.159
N (x; X,σ 2 )
9 Also called normal distribution, denoted as
9 Pdf has a single peak.
9 δ ( x − X ) = lim( 2 πσ ) − 1 exp[ − ( x − X
)2 /(2 σ
2 )]
, a good representation for a delta
0
σ
→
function because a Gaussian pdf is infinitely differentiable.
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #4

2. Gaussian Integral
1
− −
f ( x ) dx 1 exp ( x X )
X dx πσ σ
2
2
2
2

= 

∫ = ∫∞
−∞
∞
−∞
Show that
2 ∞ 2 ∞ 2 ∞
∞
2 2 = − G 2 = e − x dx e −
y dy = e − ( x +
y )
∫ −∞
∫ dxdy −∞
∫ −∞
∫ −∞
x = r cosθ and y = r sinθ
G ∫∞
e x dx −∞
Change to polar coordinates,
dxdy = rdrdθ and x2 + y2 = r2
2
0 0
∞
G 2 = e − ( x 2 + y 2 ) π
∞ dxdy = dθ e − r 2 ∞ rdr =
2∫∞ π e −
r 2 ∫ dr −∞
∫ −∞
∫ ∫ 0
Further use change of variable : t = r2 and dt = 2rdr
π π π =  
 
= ∫ = ∫∞ − ∞ −
2 2 G 2 e r rdr e tdt G = π
0 0
∞ x 2 1 ∞
−
x 2 ∫ e−dx = e dx
= 1 −∞
∫ −∞
π
π or equivalently,
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #5
Gaussian Distribution
• Often expressed in zero mean and unity variance form with
,
u x X −
σ
=

 −
x u du x ∫ −∞ 
( ) 1
2
Or equivalently, define ,
Then, F X ( x )
= Φ
x X 
 −
Q x u du x
• TheQ-function
• The error function

2
f u = u u U ,
∞ < < ∞ − 
 −

2
exp
( ) 1
2
π
2
exp
2

Φ =
π


 −

σ
1 ( )
2
exp
( ) 1
= ∫∞
2
2
x
Φ − = 

π
x [ u ]du x = ∫ −
erf ( ) 2 exp 2
0
π
Q x erf x
= −  x x
, 0
2
exp( −
/ 2)
2
a x a x b
(1 )
1
2
( ) 1

1
2
2
≥

 

 
− + +

≈ 




π
a = 0.339, b = 5.510
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #6
Functions of Random Variable
• X is a random variable with pdf fX(x).
• Y is a monotonic function of X; Y = g(X). Find fY(y).
Y = g(X )
Y
y y − Δ Y = g(X )
Positive slope,
dy/dx is positive
X
Y
y
x x x + Δ
Pr( ) Pr( ) x y x < X ≤ x + Δ = y < Y ≤ y + Δ
f x dx f y dy f y f x dx X Y Y X ( ) = ( ) or ( ) = ( )
dy
Negative slope,
dy/dx is negative
X
is negative
x
Δ
y y + Δ
y
x
x x + Δ
x + Δ < X ≤ x = y < Y ≤ y + Δ
Pr( ) Pr( )
x y
−
f X x dx f Y y dy f Y y f X
x dx dy
( ) = ( )(− ) or ( ) = ( )
f y f x dx Y X ( ) = ( )
Hence, dy
1
−
y g y x f y f g y dg y Y X
( ) , ( ) ( ( )) ( )
dy
1 1
Expressed in with − = = −
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #7
Non-Monotonic Functions of R.V.
Y = g(X )
X
Y
y y + Δ
y
1 x
1 x1 x + Δ
2 x
2 x2 x + Δ
3 x
3 x3 x + Δ
( ) ( ) ( ) 1 2 3 y = g x = g x = g x
y Y y
Pr( )
y
< ≤ + Δ
x X x
Pr( )
x
= < ≤ + Δ
1 1 1
x X x
Pr( )
+ + Δ < ≤
x
2 2 2
x X x
Pr( )
x
+ < ≤ + Δ
3 3 3
f y f g y dg y
( ) ( ( )) ( )
Σ
= −
−
1
= −
Y X 1
dy
x g y
for all ( )
1
Y = X X = Y dx = , but for any > 0, = ±
y x y
2 or , 1
dy 2
y
[ ( ) ( )], 0; 0, 0
( ) = 1 f y + f − y y ≥ = y <
f y Y X X
2
y
Example:
Therefore,
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #8, Slide #8