[PDF] - Rev eview iew of of Probabilit obability Theor heory and and PDF Document

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CMPE 252A: Computer Networks

Rev eview iew of

f Probabilit
bability

Theor heory and and Random andom Proces

cesses

es

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Probability Theory

 A mathematical model used to quantify the

likelihood of events taking place in an experiment in which events are random.

 It consists of:

 A sample space: The set of all possible

utcomes of a random experiment.

 The set of events: Subsets of the sample

space.

 The probability measure: Defined according to a

probability law for all the events of the sample space.

3

Probability Law

 Let E be a random experiment (r.e.)  Let A be an event in E  The probability of A is denoted by P(A)  A probability law for E is a rule that assigns P(A)

to A in a way that the following conditions, taken from our daily experience, are satisfied:

 A may or may not take place; it has some

likelihood (which may be 0 if it never occurs).

 Something must occur in our experiment.  If one event negates another, then the likelihood

that either occurs is the likelihood that one occurs plus the likelihood that the other occurs.

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Probability Law

 More formally, we state the same as follows.  A probability law for E is a rule that assigns a

number P(A) to each event A in E satisfying the following axioms:

 AI:  AII:  AIII:

P(A) 0 ≤ 1 ) ( = S P

P(B) P(A) B A P B A + = ⇒ = ) (   φ

Everything else is derived from these axioms!

5

Important Corollaries

1. 2. 3. 4. 5.

∑

= ⇒ ≠ ∀ Φ =

k k k k j i n

A P A P j i A A A A A ) ( ) ( ... , ,

2 1





) ( 1 ) ( A P A P

c

− =

1 ) ( ≤ A P

) ( = Φ P ) ( ) ( ) ( ) ( B A P B P A P B A P   − + =

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Probability Law

 All probabilities must be in [0, 1]  The sum of probabilities must be at most 1

A S x ) ( 1 S P = ) ( Φ = P

Φ

) (A P

) (x P

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Conditional Probability

Events of interest occur within some context, and that context changes their likelihood.

time packet collisions time interarrival times

We are interested in events occurring given that

thers take place!

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Conditional Probability

 The likelihood of an event A occurring given

that another event B occurs is smaller than the likelihood that A occurs at all.

 We define the conditional probability of A

given B as follows:

) ( for ) ( ) ( ) | ( > ∩ = B P B P B A P B A P

We require P(B) > 0 because we know B occurs!

9

Theorem of Total Probability

 Purpose is to divide and conquer  We can describe how likely an event is by

partitioning it into mutually exclusive pieces.

time

success failure

busy period idle period busy period

1

B

2

B

3

B

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Theorem of Total Probability

1

B B3

n

B

...

) ( ) ( ) | ( ) (

∑ ∑

∩ = =

i i i i i

B A P B P B A P A P

Intersections of A with B’s are mutually exclusive A

2

B A∩B2

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Independence of Events

 In many cases, an event does not depend on prior events,

r we want that to be the case.

 Example: Our probability model should not have to account

for the entire history of a LAN.

 Independence of an even with respect to another means

that its likelihood does not depend on that other event.

 A is independent of B if P(A | B) = P(A)  B is independent of A if P(B | A) = P(B)

 So the likelihood of A does not change by

knowing about B and viceversa!

 This also means:

) ( ) ( ) ( B P A P B A P = ∩

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Random Variables

 We are not interested in describing the likelihood

f every outcome of an experiment explicitly.

 We are interested in quantitative properties

associated with the outcomes of the experiment.

 Example:

 What is the probability with which each packet

sent in an experiment is received correctly? We don’t really care!

 What is the probability of receiving no packets

correctly within a period of T sec.? We care! This may make a router delete a neighbor!

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Random Variables

 We implicitly use a measurement that assigns a numerical

value to each outcome of the experiment.

 The measurement is [in theory] deterministic; based on

deterministic rules.

 The randomness of the observed values of the

measurement is completely determined by the randomness

f the experiment itself.

 A random variable X is a rule that assigns a

numerical value to each outcome of a random experiment Yes, it is really a function!

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Random Variables

 Definition: A random variable X on a sample

space S is a function X: S ->R that assigns a real number X(s) to each sample point s in S.

∞ − ∞ +

S

5

s

3

s

2

s

1

s

4

s

) ( 2 s X ) ( 1 s X ) ( ) (

5 4

s X s X =

ℜ ⊂ ∈ = } | ) ( { S s s X S

i i X

which is called the event space

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Random Variables

 Purpose is to simplify the description of the problem.  We will not have to define the sample space!

∞ + ∞ −

1

S s

x s X = ) (

) ) ( ( x s X P =

Possible values of X

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Types of Random Variables

 Discrete and continuous: Typically used

for counting packets or measuring time intervals.

time t time Busy period next packet

time?

1 2 3 4

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Discrete Random Variables

 We are interested in the probability that a discrete random

variable X (our measurement!) equals a certain value or range of values.

 Example:

We measure the delay experienced by each packet sent from one host to another over the Internet; say we sent 1M packets (we have 1M delay measurements). We want to know the likelihood with which any one packet experiences a delay of 5 ms or less.

 Probability Mass Function (pmf) of a random

variable X :

The probability that X assumes a given value x

) ( ) ( x p x X P

X

= =

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Discrete Random Variables

Cumulative Distribution Function (cdf) of a random variable X: The probability that X takes on any value in the interval ] , ( x −∞ ) ( ) ( x X P x FX ≤ =

The pdf and pmf of a random variable are just probabilities and obey the same axioms AI to

AIII. Therefore,

) ( ) ( b) X P(a for ) ( ) ( ) ( lim ; 1 ) ( lim ; 1 ) ( a F b F b a b F a F x F x F x F

X X X X X X x X

x

− = ≤ < < ≤ = = ≤ ≤

−∞ →

+∞ →

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Continuous Random Variables

 The probability that a continuous r.v. X assumes a given

value is 0.

 Therefore, we use the probability that X assumes a range

f values and make that length of that range tend to 0.

 The probability density function (pdf) of X, if it exists,

is defined in terms of the cdf of X as

dx x dF x f

X X

) ( ) ( =

∫ ∫ ∫

∞ + ∞ − ∞ −

= +∞ → ≤ ≤ ∞ − = = = ≤ ≤ y probabilit a is pdf because , 1 ) ( then if ) ( ) ( ) ( ; (x)dx f b) X P(a

b a X

dt t f x x X P dt t f x F

X x X X 20

What We Will Use

 We are interested in:  Using the definitions of well-known

r.v.s to compute probabilities

 Computing average values and

deviations from those values for well-known r.v.s

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Mean and Variance

 What is the average queue length at each

router?

 What is our worst case queue?

B A D C 3 5 4 6 1 2 7

For VC1 use 3 For VC2 use 2 ….. For VCn use 3

VC1 VC2

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Mean and Variance

Expected value or mean of X: us! for defined always is Mean c.r.v for ) ( ) ( d.r.v. for ) ( ) (

∫ ∑

∞ + ∞ −

= = dt t tf X E k p x X E

X X k k

time queue size mean too much?

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Variance

Describes how much a r.v. deviates from its average value, i.e., D = X - E(X) We are only interested in the magnitude of the deviation, so we use:

2 2

)) ( ( X E X D − =

The variance of a r.v. is defined as the mean squared variation

) (

2

D E ) )] ( ([ ) (

2 2

X E X E X Var − = = σ

Important relation:

) ( ) ( ) (

2 2

X E X E X Var − =

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Properties of Mean and Variance

Useful when we discuss amplifying random variables or adding constant biases.

) ( ) ( ) ( ) ( ) ( ) ( ; ) ( ) (

2

X Var c cX Var X Var c X Var c Var b b E b X aE b aX E = = + = = + = +

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Examples of Random Variables

 We are interested in those r.v. that permit us to

model system behavior based on the present state alone.

 We need to

 count arrivals in a time interval  count the number of times we need to repeat

something to succeed

 count the number of successes and failures  measure the time between consecutive arrivals

 The trick is to map our performance questions into

the above four types of experiments

26

Bernoulli Random Variable

 Let A be an event related to the outcomes

f a random experiment.

 X = 1 if outcome occurs and 0 otherwise  This is the Bernoulli r.v. and has two

possible outcomes: success (1) or failure (0)

q p X P P P p q X P P P

X X

− = = = = = − = = = = = 1 ) 1 ( ) 1 ( 1 ) ( ) (

1

We use it as a building block for other types of counting

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Geometric Random Variable

 Used to count the number of attempts needed to

succeed doing something.

 Example: How many times do we have to

transmit a packet over a broadcast radio channel before it is sent w/o interference?

time Assume that each attempt is independent of any prior attempt! Assume each attempt has the same probability of success (p) failure failure failure success!

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Geometric Random Variable

 We want to count the number k of trials needed for

the first success in a sequence of Bernoulli trials!

,... 2 , 1 ; ) 1 ( ) ( ) ( trial a in success

f

y probabilit

1

= − = = = =

−

k p p k X P k p p

k X

Why this is the case is a direct consequence of assuming independent Bernoulli trials, each with the same probability of success: k-1 failures needed before the last successful trial

Memoryless property: The probability of having a success in k additional trials having experienced n failures is the same as the probability of success in k trials before the n failed trails

) ( ) | ( k X P n X k n X P = = > + =

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Binomial Random Variable

 X denotes the number of times success

ccurs in n independent Bernoulli trials.

success 1 where 0); , ... 0, 0, 1, , ... 1, 1, , 1 ( such that

utcome

specific he Consider t = = ∈ s S s

k n k c n c k k

p p A P A P A P A P A P s P

− +

− = = ) 1 ( ) ( )... ( ) ( ).... ( ) ( ) ( : t independen are trials Because

1 2 1

s 0' and s 1' have we ; 1 1 n-k k n k k +

q p A P A P p A P A A A A A s i A

c i c n c k k i

= − = − = = ∩ ∩ ∩ ∩ ∩ ∩ = =

+

1 ) ( 1 ) ( ; ) ( ... ... then , in trial success Let

1 2 1

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Binomial Random Variable

k n k c n c k k

p p A P A P A P A P A P s P

− +

− = = ) 1 ( ) ( )... ( ) ( ).... ( ) ( ) (

1 2 1

! )! ( ! k n and slots, in the s 1' for positions choose to ways different k n are There k k n n n k − = " " # $ % % & ' " " # $ % % & ' Because each outcome is mutually exclusive of the others: n k p p k k n n p p k n k P

k n k k n k n

≤ ≤ − # # $ % & & ' ( − = − # # $ % & & ' ( =

− −

) 1 ( ! )! ( ! ) 1 ( ) (

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Poisson Random Variable

 We use this r.v. in cases where we need to count event

ccurrences in a time period.

 An event occurrence will typically be a packet arrival.  Arrivals are assumed to occur at random over a time

interval.

 The time interval is (0, t]  We divide the time interval into n small subintervals of

length t n t Δ = /

The probability of a new arrival in a given subinterval is

defined to be t

Δ λ

is constant, independent of the subinterval.

λ

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Poisson Random Variable

A sequence of n independent Bernoulli trials; with X being

the number of arrivals in (0, t]

arrivals. 1

r

having

f

y probabilit the to compared negligible is in arrival

ne

than more

f

y probabilit The : and Make t t n Δ → Δ ∞ →

By assumption, whether or not an event occurs in a subinterval is independent of the outcomes in other subintervals. We have:

t Δ

….

time

t

arrival 1 2 3 4 n 1 2 3 k

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Poisson Random Variable

arrivals

f

number the is ; ) 1 ( ) ( k p p k n k P

k n k n −

− " " # $ % % & ' = n t t n t t P p / } / in arrival { λ λ = Δ = = Δ =

k k n k

n t n t k k n n P k X P t k P ! " # $ % & ! " # $ % & − − = = = =

−

λ λ 1 ! )! ( ! ) ( ]} [0, in arrivals { : then , and Make → Δ ∞ → t n

k n k k k

n t n t k t k n n n P

−

" # $ % & ' − " # $ % & ' − " # $ % & ' − = λ λ λ 1 1 ! ) ( )! ( ! : get to Rearrange

[]

k n k k

n t k t n k n n n n

−

" # $ % & ' − " # $ % & ' + − − − 1 1 ! ) ( ) 1 )..( 2 )( 1 ( λ λ

P

k = n k

n t k t ! " # $ % & − λ λ 1 ! ) ( ) 1 (

n x x e

x x

− = " # $ % & ' + ≡

∞ →

with ; 1 1 lim also and

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Poisson Random Variable

 We see that the Poisson r.v. is the result of an

approximation of i.i.d. arrivals in a time interval.

 We will say “arrivals are Poisson” meaning we can use the

above formulas to describe packet arrivals

 The probability of 0 arrivals in [0, t] plays a key role in our

treatment of interarrival times.

P

k = P{k arrivals in (0,t]} = λt

( )

k

k! e−λt

t

e t P P

λ −

= = ]} (0, in arrivals {

t

e P t P

λ −

− = − = 1 1 ]} (0, in arrivals some {

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Important Properties of Poisson Arrivals

 The aggregation of Poisson sources is a Poisson

source.

 If packets from a Poisson source are routed such

that a path is chosen independently with probability p, that stream is also Poisson, with rate p times the original rate.

 It turns out that the time of a given Poisson

arrival is uniformly distributed in a time interval

The parameter λ

is called the arrival rate

36

Exponential Random Variable

 Consider the Poisson r.v., then

t

e P

λ −

=

Let Y be the time interval from a time origin (chosen

arbitrarily) to the first arival after that origin

Let

] (0, in arrivals

f

Number ) ( τ τ = X then

λτ

τ τ

−

= = = > e X P Y P ) ) ( ( ) (

We can obtain now a c.d.f. for Y as follows:

; 1 ) ( 1 ) ( ) ( ≥ − = > − = ≤ =

−

τ τ τ τ

λτ

e Y P Y P F

Y

and the p.d.f. of Y is then ; ) ( ) ( ≥ = ≤ =

−

τ λ τ τ

λτ

e Y P dt d fY

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Memoryless Property of Exponential Random Variable

 After waiting h sec. for the first arrival, the probability

that it occurs after t sec. equals the prob. that the

first arrival occurs after t sec.

 Knowing that we have waited any amount of time does

not improve our knowledge of how much longer we’ll have to wait!

) ( ) | ( h Y P t Y h t Y P > = > + >

time

time > t? time? time >t ?

38

Mean and Variance Results

You have to memorize these! [ In theory, you could be able to derive any of the above]

Exponential:

2 2

/ 1 ; / 1 ) ( λ σ λ = = Y E

Poisson:

t X E λ σ = =

2

) (

Geometric:

2 2

/ ; / 1 ) ( p q p X E = = σ

Binomial:

npq np X E = =

2

; ) ( σ

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Random Processes

 Definition: A random process is a family of

random variables {X(t) | t is in T } defined on a given sample space, and indexed by a parameter t, where t varies over an index set T.

t

x x x x x x x x x x x

1

t

2

t

3

t

4

t

5

t

a state of X(t) X(t5) is a R.V. state space of random process

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Counting Process

 A counting process N(t) starts at some time 0 and

counts the occurrences of events.

 Events are called arrivals.  Every sample function of N(t) equals 0 for all t ≤ 0.  The number of arrivals up to any time t > 0 is an

integer that cannot decrease with time. Definition: A random process N(t) is a counting process if for every sample function, n(t, s) = 0 for t < 0 and n(t, s) is integer-valued and non-decreasing with time.

Counting Process

 Jumps in the sample function of N(t) mark the arrivals.  The number of arrivals in (ti, tj ] is N(tj) – N(ti).  A simple counting process is derived from a Bernoulli

process (a sequence of i.i.d. Bernoulli random variables).

42

Counting Process

 Consider a Bernoulli process (a sequence of i.i.d. Bernoulli random

variables) with a small time step of size Δt seconds.

 There is an arrival in interval ( nΔt, (n + 1)Δt ) if and only if Xn = 1.  For an average rate λ > 0 arrivals/second we can choose Δt such

that rate λΔt << 1.

 We let the success probability of Xn be equal to λΔt.  The number of arrivals Nm before time T = mΔt (i.e., in the time

interval [0, mΔt ) ) has a binomial PMF (with λΔt = λT /m ) : P

Nm(k) =[Nm = k]=

m k ! " # $ % &(λT / m)k(1− λT / m)m−k (1)

 We have shown that, as m tends to

infinity or Δt tends to 0, Eq. (1) becomes the Poisson random variable with the PMF:

P

N (T )(k) =

(λT)k k! e−λT k = 0,1,2,...

therwise

" # $ % $

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Counting Process

IMPORTANT:

 We can generalize our argument to any interval (ti, tj ].  The number of arrivals would have a Poisson PMF with

T = tj – ti.

 The number of arrivals in T depends only on the Bernoulli

trials corresponding to that interval.

 The numbers of arrivals in non-overlapping intervals are

independent.

 In the limit, as Δt tends to 0, the number of arrivals in any

time interval is a Poisson random variable independent of the arrivals in any other interval.

 This defines the Poisson random process!

Poisson Process

Definition: A counting process N(t) is a Poisson process of rate λ if

(a) The number of arrivals in any interval (ti, tj ],

N(tj) – N(ti), is a Poisson random variable with expected value λ( tj – ti ).

(b) For any pair of non-overlapping time intervals

(ti, tj ] and (tp, tq ], the number of arrivals in (ti, tj ] and (tp, tq ] are independent random variables.

 λ is called the rate of the Poisson process, because

the expected number of arrivals per unit time is N(t) / t = λ.

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Poisson Process

 We assume that events occur at random instants of time at an

average rate of λ events per second.

 Let N(t) be the number of event occurrences (e.g., packet

arrivals) in (0,t].

 N(t) is a non-decreasing, integer-valued, continuous time random

process. t

1

a

2

a

3

a

4

a

5

a

5 4 3 2 1

a r.v. a r.v.

N(t2) N(t1)

We have a Poisson random variable for each time instant!

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46

Poisson Process

 As we did for the case of the Poisson r.v., we divide the

time axis into very small subintervals, such that

 The probability of more than one arrival in a subinterval is much

smaller than the probability of 0 or 1 arrivals in the subinterval.

 Whether or not an arrival occurs in a subinterval is independent of

what takes place in any other subinterval.

 We end up with the same Binomial counting process and

with the length of the subintervals going to 0 we can approximate:

( )

,... 1 , , ! ]} (0, in arrivals { ] ) ( [ = = = =

−

k e k t t k P k t N P

t k λ

λ

Again, inter arrival times are exponentially distributed with parameter lambda

47

Poisson Process

 Why do we say that arrivals occur “at random” with a Poisson

process?

 Suppose that we know that only one arrival occurs in [0, t] and x

is the arrival time of the single arrival.

 Let N(x) be the number of arrivals up to time x, 0 ≤ x ≤ t  Then:

N(t) – N(x) = increment of arrivals in ( x, t ] } 1 ) ( { } ) ( ) ( and 1 ) ( { } 1 ) ( { } 1 ) ( ) ( { } 1 ) ( | 1 ) ( { ] [ = = − = = = = = = = = = ≤ t N P x N t N x N P t N P t N x N P t N x N P x X P

48

Poisson Process

 Arrivals in different intervals are independent.

} 1 ) ( { } ) ( ) ( P{ } 1 ) ( { ] [ = = − = = ≤ t N P x N t N x N P x X P t x te e x x X P

t x

= = ≤

− − λ λ λ

λ λ ] ][e ) [( ] [

x)

(t
The above probability corresponds to the