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Discrete Random Variables

Randomness

• The word

random

effectively means

unpredictable

• In engineering practice we may treat some

randomeven though they may not actually besignals as random to simplify the analysis

Random Variables

experiments. Examples of assignments of numbers to the outcomes of

Discrete-Value vs Continuous-

Value Random Variables

A

discrete-value (DV)

random variable has a set

occurof distinct values separated by values that cannot

be DV random variableof coin flips, card draws, dice tosses, etc... wouldA random variable associated with the outcomes

A

continuous-value (CV)

random variable may

may be finite or infinite in sizetake on any value in a continuum of values which

A DV random variable

X

is a

(^) Bernoulli

random variable if it

takes on only two values 0 and 1 and its pmf is

P

X

x

p

, x (^) = (^0)

p

x (^) = (^1)

, otherwise

and 0

p (^) <

(^) 1.

Probability Mass Functions

Example of a Bernoulli pmf

Probability Mass Functions

Binomial pmf

Probability Mass Functions

probability of a 1 on any single trial isIf we perform Bernoulli trials until a 1 (success) occurs and the

(^) p , the probability that the

first success will occur on the

(^) k th trial is

(^) p

1
p

k
1

. A DV random

variable

X

is said to be a

(^) Geometric

random variable if its pmf is

P

X

x

p

1
p

( ) x
1 , x

, otherwise

Probability Mass Functions

If we perform Bernoulli trials until the

(^) r th 1 occurs and the

probability of a 1 on any single trial is

(^) p , the probability that the

r th success will occur on the

(^) k th trial is

P

r th success on

(^) k th trial

k (^)
(^1)

r (^)
(^1)

p (^) r 1
p

( ) k
r.

A DV random variable

Y

is said to be a

(^) negative - Binomial

or (^) Pascal

random variable with parameters

(^) r

and

(^) p

if its pmf is

P

Y

y

y (^)
(^1)

r (^)
(^1)

p (^) r 1
p

( ) y
r , y

r , r (^) + (^) 1,

, 

, otherwise

Probability Mass Functions

(Pascal) pmfNegative Binomial

Probability Mass Functions

Events occurring at random times

Probability Mass Functions

It can be shown that

P

k inside

t  (^) =

(^) k

k ! n lim 

n 

n

= e


(^) k

k ! (^) e


where

t

. A DV random variable is a Poisson random

variable with parameter

if its pmf is

P

X

x

(^) x

x ! (^) e
 , x (^) 

, otherwise

Probability Mass Functions

Consider a transformation from a DV random variableFunctions of a Random Variable

X

to another DV random variable

Y

through

Y

(^) g

X

. If the

function g is invertible, then

X

(^) g
1 Y

and the pmf for

Y

is

P

Y

y

P

X

g
1

y

where P

X

x

is the pmf for X.

Functions of a Random Variable

If the function g is not invertible the pmf and pdf of

Y

can be found

by finding the probability of each value of

Y

. Each value of

X

with

corresponding value ofnon-zero probability causes a non-zero probability for the

Y

. So, for the

(^) i th value of

Y

P

Y

y i

P

X

x i ,^

P

X

(^) x i ,

P

X

x i ,^ n

P

X

x i , k

k

1

n

function.example of a non-invertible The function to the right is an