Chapter 5 Discrete Probability Distributions
... waiting in line. However, arrivals are a random event. If we consider the two minute window for servicing a customer as our “bucket”, we can ask, what is the probability that we’ll end up with people waiting one minute, two minutes, three minutes, etc. 2. Let’s say that a monkey is throwing darts at ...
... waiting in line. However, arrivals are a random event. If we consider the two minute window for servicing a customer as our “bucket”, we can ask, what is the probability that we’ll end up with people waiting one minute, two minutes, three minutes, etc. 2. Let’s say that a monkey is throwing darts at ...
Poisson Processes and Applications in Hockey
... on previous events. If scoring in hockey is not memoryless then goals would occur in bunches similar to baseball. We can see that goals in hockey are indeed rare, memoryless and for the most part they are random, the exception is during the final minutes of the third period when teams are trailing b ...
... on previous events. If scoring in hockey is not memoryless then goals would occur in bunches similar to baseball. We can see that goals in hockey are indeed rare, memoryless and for the most part they are random, the exception is during the final minutes of the third period when teams are trailing b ...
Bayesian Networks without Tears
... An Example Bayesian Network The best way to understand Bayesian networks is to imagine trying to model a situation in which causality plays a role but where our understanding of what is actually going on is incomplete, so we need to describe things probabilistically. Suppose when I go home at night, ...
... An Example Bayesian Network The best way to understand Bayesian networks is to imagine trying to model a situation in which causality plays a role but where our understanding of what is actually going on is incomplete, so we need to describe things probabilistically. Suppose when I go home at night, ...
On solutions of stochastic differential equations with parameters
... but only for almost all ω, that is, for some subset of Ω whose probability is 1. That is why the term version is frequently used. Two stochastic processes x and x̃ are called versions of each other (or stochastically equivalent) if for all t ∈ T it holds that P ({ω : xt (ω) = x̃t (ω)}) = 1. The firs ...
... but only for almost all ω, that is, for some subset of Ω whose probability is 1. That is why the term version is frequently used. Two stochastic processes x and x̃ are called versions of each other (or stochastically equivalent) if for all t ∈ T it holds that P ({ω : xt (ω) = x̃t (ω)}) = 1. The firs ...
Bayesian Networks without Tears
... An Example Bayesian Network The best way to understand Bayesian networks is to imagine trying to model a situation in which causality plays a role but where our understanding of what is actually going on is incomplete, so we need to describe things probabilistically. Suppose when I go home at night, ...
... An Example Bayesian Network The best way to understand Bayesian networks is to imagine trying to model a situation in which causality plays a role but where our understanding of what is actually going on is incomplete, so we need to describe things probabilistically. Suppose when I go home at night, ...
Stochasticity, invasions, and branching random walks
... Missing from this recent work is a clear and complete outline of the linear, density-independent theory (but see Lewis and Pacala, 2000). A linear theory provides us with a useful basis for judging nonlinear effects. It may help us distinguish the effects of stochasticity from the complicated effect ...
... Missing from this recent work is a clear and complete outline of the linear, density-independent theory (but see Lewis and Pacala, 2000). A linear theory provides us with a useful basis for judging nonlinear effects. It may help us distinguish the effects of stochasticity from the complicated effect ...
Scalable Analysis and Design of Ad Hoc Networks Via Random
... As a general comment, it is worth mentioning that an RVM random graph, in full generality, can mimic any possible random graph model on a given number of vertices, as it allows any probability distribution over the set of all graphs on a given number of vertices. To see this let G1 , . . . , GN be a ...
... As a general comment, it is worth mentioning that an RVM random graph, in full generality, can mimic any possible random graph model on a given number of vertices, as it allows any probability distribution over the set of all graphs on a given number of vertices. To see this let G1 , . . . , GN be a ...
ECS 455: Mobile Communications Call Blocking Probability
... both of which estimate the call blocking probability when trunking is used. To do this, we need to borrow some concepts from queueing theory. Moreover, some basic analysis of stochastic processes including Poisson processes and Markov chains is needed. For completeness, working knowledge on these pr ...
... both of which estimate the call blocking probability when trunking is used. To do this, we need to borrow some concepts from queueing theory. Moreover, some basic analysis of stochastic processes including Poisson processes and Markov chains is needed. For completeness, working knowledge on these pr ...
Lecture 3: Continuous times Markov chains. Poisson Process. Birth
... parameter λi h so that λi is the expected number of birth events that occur per unit time. In this case, the probability of a birth over a short interval h is λi h + o(h). Similarly, if in state X(t) = i a death rate is µi , then the probability that an individual dies in a very small time interval ...
... parameter λi h so that λi is the expected number of birth events that occur per unit time. In this case, the probability of a birth over a short interval h is λi h + o(h). Similarly, if in state X(t) = i a death rate is µi , then the probability that an individual dies in a very small time interval ...
Creating Probability Models for Simple Events
... Clarifying the Standards Prior Learning In earlier grades, students developed knowledge and experience with data. They viewed statistical reasoning as a four step process that included: formulating questions that could be answered using data, designing and using a plan to collect relevant data, anal ...
... Clarifying the Standards Prior Learning In earlier grades, students developed knowledge and experience with data. They viewed statistical reasoning as a four step process that included: formulating questions that could be answered using data, designing and using a plan to collect relevant data, anal ...
Document
... • The queue discipline describes the method used to determine the order in which customers are served. • The most common queue discipline is the FCFS discipline (first come, first served), in which customers are served in the order of their arrival. • Under the LCFS discipline (last come, first serv ...
... • The queue discipline describes the method used to determine the order in which customers are served. • The most common queue discipline is the FCFS discipline (first come, first served), in which customers are served in the order of their arrival. • Under the LCFS discipline (last come, first serv ...
Stochastic Processes
... process in probability theory a process involving the operation of chance for example in radioactive decay every atom is subject to a fixed probability, stochastic process encyclopedia of mathematics - where is an arbitrary dimensional vector therefore the study of one dimensional processes occupies ...
... process in probability theory a process involving the operation of chance for example in radioactive decay every atom is subject to a fixed probability, stochastic process encyclopedia of mathematics - where is an arbitrary dimensional vector therefore the study of one dimensional processes occupies ...
... Discrete ancestral problems arising in population genetics are investigated. In the neutral case, the duality concept has been proved of particular interest in the understanding of backward in time ancestral process from the forward in time branching population dynamics. We show that duality formula ...
Using Metrics in Stability of Stochastic Programming Problems
... easily seen, usual real-life tasks cannot meet this assumption: the probability distribution is not really known but only estimated, for example, from historical data series. In another class of models the probability distribution is known but far more complicated than we could solve the problem eas ...
... easily seen, usual real-life tasks cannot meet this assumption: the probability distribution is not really known but only estimated, for example, from historical data series. In another class of models the probability distribution is known but far more complicated than we could solve the problem eas ...
Information Theory and Predictability. Lecture 3: Stochastic Processes
... In general if some of the transition functions are zero then it will not be possible to pass from a particular state to another in one time step. However it may be possible to do so in more than one steps i.e. indirectly. This motivates the following properties of some Markov processes: Definition 4 ...
... In general if some of the transition functions are zero then it will not be possible to pass from a particular state to another in one time step. However it may be possible to do so in more than one steps i.e. indirectly. This motivates the following properties of some Markov processes: Definition 4 ...
Carrying Capacity and Demographic Stochasticity: Scaling Behavior
... To evaluate the probability of extinction of an established population, consider the population as performing a biased random walk. Whenever the population is less than the carrying capacity, the birth rate will be greater than the death rate; thus the random walk will be biased towards increasing p ...
... To evaluate the probability of extinction of an established population, consider the population as performing a biased random walk. Whenever the population is less than the carrying capacity, the birth rate will be greater than the death rate; thus the random walk will be biased towards increasing p ...
paper
... may be onerous and inefficient. Recent work in this area signal and W ( n )is AWGN with variance u 2 .Our coding has considered the case where the SCSI is a deterministic theorem follows. function of the RCSI [l]. In [l], exact capacity results are given for the case when the SCSI remains Markov. If ...
... may be onerous and inefficient. Recent work in this area signal and W ( n )is AWGN with variance u 2 .Our coding has considered the case where the SCSI is a deterministic theorem follows. function of the RCSI [l]. In [l], exact capacity results are given for the case when the SCSI remains Markov. If ...
Modeling Data Dissemination in Online Social Networks: A
... Recent years have witnessed a dramatic growth of user population of online social networks (OSNs). For example, according to the report in March 2013, Facebook has 1.11 billion people using the site each month, which represents a 23 percent growth from a year earlier [3]. OSNs are organized around u ...
... Recent years have witnessed a dramatic growth of user population of online social networks (OSNs). For example, according to the report in March 2013, Facebook has 1.11 billion people using the site each month, which represents a 23 percent growth from a year earlier [3]. OSNs are organized around u ...
pdf 160k - Ray Solomonoff
... acceptable sentences in some simple formal language. We are required to find a grammar that could generate these strings. In general, there will be an infinite number of grammars that can generate the set even if we restrict the grammars, say, to be finite state or to be context free grammars. Early ...
... acceptable sentences in some simple formal language. We are required to find a grammar that could generate these strings. In general, there will be an infinite number of grammars that can generate the set even if we restrict the grammars, say, to be finite state or to be context free grammars. Early ...
Exam 1 Solution 1. (10 pts) The following circuit operates if and only
... 5. (10 pts) It is conjectured that an impurity exists in 30% of all drinking wells in a certain rural community. In order to gain some insight on this problem, it is determined that some tests should be made. It is too expensive to test all of the many wells in the area, so 10 were randomly selected ...
... 5. (10 pts) It is conjectured that an impurity exists in 30% of all drinking wells in a certain rural community. In order to gain some insight on this problem, it is determined that some tests should be made. It is too expensive to test all of the many wells in the area, so 10 were randomly selected ...
Bayesian Networks and Hidden Markov Models
... In a Hidden Markov Model (HMM) the a Markov Chain is expanded to include the idea of hidden states. Given a set of observations x1, x2…xn and a set of hidden underlying states s1, s2…sn, there is now a transition probability for moving between the hidden states: ...
... In a Hidden Markov Model (HMM) the a Markov Chain is expanded to include the idea of hidden states. Given a set of observations x1, x2…xn and a set of hidden underlying states s1, s2…sn, there is now a transition probability for moving between the hidden states: ...
STOCHASTIC PROCESSES Basic notions
... A single state which alone forms a closed set is called an absorbing state - for an absorbing state we have pi,i = 1 - one may reach an absorbing state from other states, but one cannot get out of it Each state is either transient or recurrent. • A state i is transient if the probability of returnin ...
... A single state which alone forms a closed set is called an absorbing state - for an absorbing state we have pi,i = 1 - one may reach an absorbing state from other states, but one cannot get out of it Each state is either transient or recurrent. • A state i is transient if the probability of returnin ...
Error Metrics for Business Process Models
... human modelers loose track of the interrelations of large and complex models due to their limited cognitive capabilities, and then introduce errors that they would not insert in a small model. A recent study provides first evidence for this hypothesis [2]. Before we can test such a hypothesis appropr ...
... human modelers loose track of the interrelations of large and complex models due to their limited cognitive capabilities, and then introduce errors that they would not insert in a small model. A recent study provides first evidence for this hypothesis [2]. Before we can test such a hypothesis appropr ...
Some discrete distributions
... Let X be an integer in the range 10..20. P(x) = 1/11 for x = 10,11,12,13,14,..20 To find E(X): Let Y be a number in the range 1..11. Thus, E(X) = (1+11)/2 = 6. Now, X = Y + 9. E(X) = E(Y+9) = E(Y) + 9 = 6+9 = 15 Var(X) = Var(Y+9) = Var(Y) = (112 – 1)/12 = 10 Note: the Discrete Uniform Distribution i ...
... Let X be an integer in the range 10..20. P(x) = 1/11 for x = 10,11,12,13,14,..20 To find E(X): Let Y be a number in the range 1..11. Thus, E(X) = (1+11)/2 = 6. Now, X = Y + 9. E(X) = E(Y+9) = E(Y) + 9 = 6+9 = 15 Var(X) = Var(Y+9) = Var(Y) = (112 – 1)/12 = 10 Note: the Discrete Uniform Distribution i ...
Slides - RAD Lab - University of California, Berkeley
... – mathematically this simply means that we work with stochastic processes ...
... – mathematically this simply means that we work with stochastic processes ...
Stochastic geometry models of wireless networks
In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.In the early 1960s a pioneering stochastic geometry model was developed to study wireless networks. This model is considered to be the origin of continuum percolation. Network models based on geometric probability were later proposed and used in the late 1970s and continued throughout the 1980s for examining packet radio networks. Later their use increased significantly for studying a number of wireless network technologies including mobile ad hoc networks, sensor networks, vehicular ad hoc networks, cognitive radio networks and several types of cellular networks, such as heterogeneous cellular networks. Key performance and quality of service quantities are often based on concepts from information theory such as the signal-to-interference-plus-noise ratio, which forms the mathematical basis for defining network connectivity and coverage.The principal idea underlying the research of these stochastic geometry models, also known as random spatial models, is that it is best to assume that the locations of nodes or the network structure and the aforementioned quantities are random in nature due to the size and unpredictability of users in wireless networks. The use of stochastic geometry can then allow for the derivation of closed-form or semi-closed-form expressions for these quantities without resorting to simulation methods or (possibly intractable or inaccurate) deterministic models.