Exponential and Erlang Distributions: Properties and Examples, Study notes of Electrical and Electronics Engineering

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RANDOM VARIABLES

OUTLINE

Other PDFs

Exponential and Erlang

Delta

Problem Solving Examples

Reading:

G. R. Cooper & C. D. McGillem 2.6 - 2.

EE/STAT 322, #

EXPONENTIAL AND RELATED DISTRIBUTIONS

The PDF is given by

f X

(^) ( x ) =

τ 1 (^) e −

τ^1 (^) x

x

otherwise

E

X

τ (^) , and

σ x 2

=

τ (^2) .

E

X

∞ 0

f (^) ( x ) xdx

∞ 0

τ 1 (^) e

−

τ^1 (^) x

xdx

By integrating by parts (

xdy

xy

ydx

), we get

E

X

∞

0

xde

−

τ^1 (^) x

xe

−

τ^1 (^) x | 0 ∞

∞

0

e −

τ^1 (^) x

dx

τ

EE/STAT 322, #

EXPONENTIAL DISTRIBUTION (CONT.)

Example:

The waiting time of a customer has an exponential distribution

with a mean of 5 minutes.

Then what is the probability that he will wait

Solution:more than 10 minutes?

τ

Pr(

X >

e −

10

/ τ = e − 2

another 10 minutes is given byIf he already spent five minutes there, the chance that he needs to wait

Pr(

X >

X >

e −

15

/ τ /e^

−

5 / τ

= Pr(

X >

EE/STAT 322, #

ERLANG DISTRIBUTION

Exponential distribution:

arrival interval between the first and second

Erlang: arrival interval between the first and the fourth customers (fig. b).customers (fig. a);

t

-- events

x

t

x

(a)

(b)

EE/STAT 322, #

DELTA DISTRIBUTION

PDF:

f (^) ( x ) =

p 1 δ ( x − x 1

p 2 δ ( x − x 2 ) ,

where

p 1

p 2

= 1

, and

δ ()

is the Kronecker delta function.

x Example: Two possible outcomes of a coin experiment. 1

= 0

for H, and

x 2

= 1

for T, and

p 1

=

p 2

= 0

E

X

X

∞ −∞

x [ p 1 δ ( x − x 1

p 2 δ ( x − x 2

)]

dx

= p 1 x 1 + p 2 x 2.

E

X

2 ) =

∞ −∞

x 2 [ p 1 δ ( x − x 1

p 2 δ ( x − x 2

)]

dx

p 1 x 12

p 2 x 22 .

σ X 2

= E ( X 2 ) − X 2 = p 1 p 2 ( x 1 − x 2 ) 2.

EE/STAT 322, #

DELTA DISTRIBUTION (CONT.)

f Extension: Multiple outcomes. (^) ( x ) =

i n

p i δ ( x − x i )

, where

i n

(^) p

i = 1

X

i n

p i x i , X 2 = ∑

i n

p i x i 2 (^) ,

σ X 2

2 1

∑

i n

j n

(^) p

i p j ( x i − x j ) 2.

Example:

(Ex 2-7.3)

Three coins are tossed. Let

H

number of heads

Find

E ( H ) , σ

H 2

(^).

EE/STAT 322, #

EXAMPLES FOR DENSITY FUNCTIONS

Example:

(Problem 2-4.5, textbook) RV

X

has a pdf of the form

f X

(^) ( x )

ax

2

< x

ax

< x

Find (a) the value of

a , (b)

E

X

, (c)

Pr(

< X

(a) We need Solution:

3 0

f X

(^) ( x ) dx

, so that

2 0

ax

2 dx

3 2

ax

ax

3 / 3 | 02

ax

2 / 2 | 23

=

a [

/ 3 + 9

2] = 1

a

(b)

E

X

2 0

xax

2 dx

3 2

xaxdx

6 31

[4 + 9

3] = 2

(c)

Pr(

< X

3 2

f X

(^) ( x ) dx

3 2

6 31

xdx

EE/STAT 322, #

EXAMPLES FOR DENSITY FUNCTIONS (CONT.)

Example:

(2-5.3) Gaussian RV

X

has a probability of 0.5 of having value

less than or equal to 1. Further,

Pr(

X >

Find (a)

E

X

; (b)

σ x 2 ; (c)

Pr(

X

(a) Solution:

Pr(

X

. So

E

X

(b)

Pr(

X >

/σ

x ) =

Q

/σ

x ) = 0

Using the inverse of

Q

-function, we get

/σ

x

= 2

, so

σ x 2

= 4

(c)

Pr(

X

/σ

x ) = Φ(1) = 0

. Alternatively,

Pr(

X

Pr(

X >

Q

/σ

x ) = 1

Q

EE/STAT 322, #

11

Example:

(2-6.3) A current

I

with a Rayleigh PDF passes through a

resistor with

R

π )Ω

E

I

A. Power dissipation:

W

RI

2 .

(a) Find the mean of power dissipated

E

W

. (b) Find

Pr(

W

. (c)

Pr(

W >

(a) Solution:

E

I

π/

σ

σ

/π

E

W

RE

I

2 ) =

R

σ

2 ) = 2

π 2 · 4 · 2

/π

(b)

Pr(

W

1. = Pr(

RI

2

≤

1. = Pr(

I

2

<

1. = Pr(

I <

F

I (^) (

exp(

1 . 382

2

2 σ 2

(c)

Pr(

W >

1. = Pr(

I >

F

I (^) (

EE/STAT 322, #

EXAMPLES FOR DENSITY FUNCTIONS (CONT.)

Example:

RV

θ

is uniformly distributed in

π ) .

Another RV

X

is given by

X

= cos(

θ ) .

(a) Find PDF of

X

; (b) Find

E

X

; (c) Find

σ x 2 ; (d) Find

Pr(

X >

(a) Solution:

dx/dθ

(^) sin

(^) θ

cos

2 θ = − √ 1 − x 2.

EE/STAT 322, #

EXAMPLES FOR DENSITY FUNCTIONS (CONT.)

At both

θ 1

and

θ 2 , | dx/dθ

| = √ 1 − x 2.

f X

(^) ( x ) =

f (^) ( θ )

√ 1 − x 2 = 1

π √ 1 − x 2 − 1 ≤ x ≤ 1

elsewhere

(b)

E

X

E

[cos(

θ )] =

2 π

0

cos(

θ )

1

2 π (^) dθ

(c)

E

X

2 ) =

X

2

=

E

[cos

2 θ ] =

2 π

0

cos

2 ( θ )

1

2 π (^) dθ

2 1 .

σ x 2

= X 2 − X 2 = X 2 =

2 1 .

EE/STAT 322, #

EXAMPLES FOR DENSITY FUNCTIONS (CONT.)

(d)

Pr(

X >

1 0 . 5 f X

(^) ( x ) dx

1 0 . 5

1

π √ 1 − x 2

(^) dx

Procedure: Let

x

= cos(

θ ) , for

θ

∈

, π/

So

dx

sin

θdθ

1

√ 1 − x 2 = 1

sin

(^) θ (^).

x

θ

= cos

−

1 (1) = 0

x

θ

= cos

− 1 (

π/

1

0 . 5

π √ 1 − x 2

(^) dx

0

π/

3

π

sin

(^) θ

( −

(^) sin

(^) θ

) dθ

π/

3

0

π 1

dθ

EE/STAT 322, #