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# Probability Review - PowerPoint PPT Presentation

Probability Review. Georgios Varsamopoulos School of Computing and Informatics Arizona State University. Problem 1. We roll a regular dice and we toss a coin as many times as the roll indicates. What is the probability that we get no tails? What is the probability that we get 3 heads?

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### Probability Review

Georgios VarsamopoulosSchool of Computing and InformaticsArizona State University

• We roll a regular dice and we toss a coin as many times as the roll indicates.

• What is the probability that we get no tails?

• What is the probability that we get 3 heads?

• Using the same process, we are asked to get at least 6 tails in total.

• If we don’t get 6 tails with the first roll, we roll again and repeat as many times as needed to get at least 6 tails.

• What is the probability that we need more than 1 rolls to get 6 tails?

• A process has to send 1000 bytes over a wireless link using a stop-and-wait protocol. The payload size per packet is 10 bytes. The bit-wise error probability is 10-3.

• Find the expected number of total packet transmissions needed to transfer the 1000 bytes.

### Markov Chains

• Discrete time Markov Chain: is a process that changes states at discrete times (steps). Each change in the state is an event.

• The probability of the next state depends solely on the current state, and not on any of the past states:

• Markov Process’s versatility

• can be used in building mobility models, data traffic models, call pattern models etc.

• Putting all the transition probabilities together, we get a transition probability matrix:

• The sum of probabilities across each row has to be 1.

• We denote Pn,n+1 as P(n). If for each step n the transition probability matrix does not change, then we denote all matrices P(n) as P.

• The transition probabilities after 2 steps are given by PP=P2; transition probabilities after 3 steps by P3, etc.

• Usually, limnPn exists, and shows the probabilities of being in a state after a really long period of time.

• Stationary probabilities are independent of the initial state

• No matter where you start, after a long period of time, the probability of being in a state will not depend on your initial state.

r

r

1

1

q

q

q

1

2

3

4

5

p

p

p

Markov Random Walk

• A Random Walk Is a subcase of Markov chains.

• The transition probabilitymatrix looks like this:

• Hitting probability (gambler’s ruin) ui=Pr{XT=0|Xo=i}

• Mean hitting time (soujourn time) vi=E[T|Xo=k]

D

Example Problem

• A mouse walks equally likely between adjacent rooms in the following maze:

• Once it reaches room D, it finds food and stays there indefinitely.

• What is the probability that it reaches room D within 5 steps, starting in A?

• Is there a probability it will never reach room D?

0.99

LOW

HIGH

0.9

0.1

Example Problem 2

• A variable bit rate (VBR) audio stream follows a bursty traffic model described by the following two-state Markov chain:

• In the LOW state it has a CBR of 50Kbps and in the HIGH it has a CBR of 150Kbps.

• What is the average bit-rate of the stream?

• We find the stationary probabilities

• Using the stationary probabilities we find the average bitrate:

• Simple variation of the time-discrete markov chain

• Self-transitions, i.e. (n,n) transitions, are replaced with the PDF of the pause.

• At any state, the process remains for a random time under exponential distribution

### Mobility Models for Mobile Computing

Georgios VarsamopoulosSchool of Computing and InformaticsArizona State University

Location management

Routing

Location-aware applications

Mobility models are used in three phases in developing a LM scheme

At assumptions phase

At performance analysis phase

At simulation phase

Analytical vs simulation mobility models

two types of mobility models:

Aggregate (analytical)

per-user (simulation)

Two granularity domains of mobility models:

Discrete space

Continuous space

Aggregate mobility models

flow models

per-user mobility models

based on a stochastic modeling of velocity.

random way-point (RWP)

Angular RWP

Habitual models

Trip-based mobility model

Markovian models

Variations

Time-dependent behavior (temporal dependency)

Location-dependent behavior (spatial dependency)

Mobility models

• First appeared in the publication of DSR[1996]

• Basic configuration

• Choose a random target point (waypoint) distributed (usually uniformly) over some area

• Move from current location to the target waypoint using a constant velocity

0

2

vo

vo

vo

1

3

• Non-uniform distribution of waypoint selection

• Uniform distribution makes the nodes spend more time around the center

• Pause times

• Select a random or a constant time to wait at each destination

• Random velocities

• Choose a random velocity and move from current location to the target waypoint using this velocity

0

2

vo

v1

v2

1

3

• Choose a random direction (way) distributed (usually uniformly) over [0, 2π]

• Choose a random velocity and move from current location over specified direction for certain time or distance.

0

1

2

v1

3

v2

2

v3

1

3

Markovian:Random Graph Mobility Process

• A random walk mobility model Gu for a user u:

• Nodes represent regions

• Weighted directed edge

• an edge r1-> r2 has weight: Pr[ user will mover to r2 when it leaves r1]

• Pr[path p in Gu] = product of probabilities of all edges in p.

• In general, Gu has cycles

P(n,m)(n,m+1)

P(n,m)(n+1,m)

Grid-based markovian mobility model(2-d random walk)

• Based on a time-continuous, space-discrete 2-D markov chain

• Can be used to model urban mobility

P(n,m)(n,m+1)

Hex-based markovian mobility model

• Variation of the 2-d model

• Can be used in cellular systems

### Probability Review

Georgios VarsamopoulosSchool of Computing and InformaticsArizona State University

### Basic Probability concepts

• Basic concepts by example: dice rolling

• (Intuitively) coin tossing and dice rolling are Stochastic Processes

• The outcome of a dice rolling is an Event

• The values of the outcomes can be expressed as a Random Variable, which assumes a new value with each event.

• The likelihood of an event (or instance) of a random variable is expressed as a real number in [0,1].

• Dice: Random variable

• instances or values: 1,2,3,4,5,6

• Events: Dice=1, Dice=2, Dice=3, Dice=4, Dice=5, Dice=6

• Pr{Dice=1}=0.1666

• Pr{Dice=2}=0.1666

• The outcome of the dice is independent of any previous rolls (given that the constitution of the dice is not altered)

• The sum of probabilities of all possible values of a random variable is exactly 1.

• The probability of the union of two events of the same variable is the sum of the separate probabilities of the events

• Pr{ Dice=1  Dice=2 } = Pr{Dice=1}+{Dice=2} = 1/3

• The tossing of two or more coins (such that each does not touch any of the other(s) ) simulateously is called (statistically) independent processes (so are the related variables and events).

• The probability of the intersection of two or more independent events is the product of the separate probabilities:

• The likelihood of an event can change if the knowledge and occurrence of another event exists.

• Notation:

• Usually we use conditional probabilities like this:

• Problem statement (Dice picking):

• There are 3 identical sacks with colored dice. Sack A has 1 red and 4 green dice, sack B has 2 red and 3 green dice and sack C has 3 red and 2 green dice. We choose 1 sack equally randomly and pull a dice while blind-folded. What is the probability that we chose a red dice?

• Thinking:

• If we pick sack A, there is 0.2 probability that we get a red dice

• If we pick sack B, there is 0.4 probability that we get a red dice

• If we pick sack C, there is 0.6 probability that we get a red dice

• For each sack, there is 0.333 probability that we choose that sack

• Result ( R stands for “picking red”):

• mth moment:

• Expected value is the 1st moment: X =E[X]

• mth central moment:

• 2nd central moment is called variance (Var[X],2X)

### Discrete Distributions

• Based on random variables that can have 2 outcomes (A and A): Pr{A}, Pr{A}=1-Pr{A}

• Expresses the probability of number of trials needed to obtain the event A.

• Example: what is the probability that we need k dice rolls in order to obtain a 6?

• Formula: (for the dice example, p=1/6)

• A wireless network protocol uses a stop-and-wait transmission policy. Each packet has a probability pE of being corrupted or lost.

• What is the probability that the protocol will need 3 transmissions to send a packet successfully?

• Solution: 3 transmissions is 2 failures and 1 success, therefore:

• What is the average number of transmissions needed per packet?

• Expresses the probability of some events occurring out of a larger set of events

• Example: we roll the dice n times. What is the probability that we get k 6’s?

• Formula: (for the dice example, p=1/6)

• Every packet has n bits. There is a probability pΒthat a bit gets corrupted.

• What is the probability that a packet has exactly 1 corrupted bit?

• What is the probability that a packet is not corrupted?

• What is the probability that a packet is corrupted?

### Continuous Distributions

• Continuous Distributions have:

• Probability Density function

• Probability Distribution function:

• Normal Distribution:

• Uniform Distribution:

• Exponential Distribution:

### Poisson Process

• Poisson Distribution is the limit of Binomial when n and p 0, while the product np is constant. In such a case, assessing or using the probability of a specific event makes little sense, yet it makes sense to use the “rate” of occurrence (λ=np).

• Example: if the average number of falling stars observed every night is λ then what is the probability that we observe k stars fall in a night?

• Formula:

• In a bank branch, a rechargeable Bluetooth device sends a “ping” packet to a computer every time a customer enters the door.

• Customers arrive with Poisson distribution of  customers per day.

• The Bluetooth device has a battery capacity of m Joules. Every packet takes n Joules, therefore the device can send u=m/n packets before it runs out of battery.

• Assuming that the device starts fully charged in the morning, what is the probability that it runs out of energy by the end of the day?

• It is an extension to the Poisson distribution in time

• It expresses the number of events in a period of time given the rate of occurrence.

• Formula

• A Poisson process with parameter  has expected value  and variance .

• The time intervals between two events of a Poisson process have an exponential distribution (with mean 1/).

• An interrupt service unit takes to sec to service an interrupt before it can accept a new one. Interrupt arrivals follow a Poisson process with an average of  interrupts/sec.

• What is the probability that an interrupt is lost?

• An interrupt is lost if 2 or more interrupts arrive within a period of to sec.