Assignment 8

... 1. Let {N (t) : t ≥ 0} be a Poisson process of rate λ. For 0 ≤ u ≤ t and n = 1, 2, . . ., determine the conditional probability of N (u) given that N (t) = n. 2. Let {N (t) : t ≥ 0} be a Poisson process of rate λ. Given that N (t) = 3, determine the conditional (marginal) probability density functio ...

... 1. Let {N (t) : t ≥ 0} be a Poisson process of rate λ. For 0 ≤ u ≤ t and n = 1, 2, . . ., determine the conditional probability of N (u) given that N (t) = n. 2. Let {N (t) : t ≥ 0} be a Poisson process of rate λ. Given that N (t) = 3, determine the conditional (marginal) probability density functio ...

Binomial random variables

... 3. The probability is 0.04 that a person reached on a “cold call” by a telemarketer will make a purchase. If the telemarketer calls 40 people, what is the probability that at least one sale with ...

... 3. The probability is 0.04 that a person reached on a “cold call” by a telemarketer will make a purchase. If the telemarketer calls 40 people, what is the probability that at least one sale with ...

1 Poisson Processes

... An arrival is simply an occurance of some event — like a phone call, a job oﬀer, or whatever — that happens at a particular point in time. We want to talk about a class of continuous time stochastic processes called arrival processes that describe when and how these events occur. Let Ω be a sample s ...

... An arrival is simply an occurance of some event — like a phone call, a job oﬀer, or whatever — that happens at a particular point in time. We want to talk about a class of continuous time stochastic processes called arrival processes that describe when and how these events occur. Let Ω be a sample s ...

The Poisson Probability Distribution

... roam in an enclosed area of 1000 square feet. The area is divided into 200 subsections of 5 square feet each. a) Assuming that the beetles spread evenly throughout the enclosed area, how many beetles are expected to be within each subsection? b) What is the standard deviation of x, the number of bee ...

... roam in an enclosed area of 1000 square feet. The area is divided into 200 subsections of 5 square feet each. a) Assuming that the beetles spread evenly throughout the enclosed area, how many beetles are expected to be within each subsection? b) What is the standard deviation of x, the number of bee ...

ECE 541 Probability Theory and Stochastic Processes Fall 2014

... distributions, distribution functions and densities; independence. ...

... distributions, distribution functions and densities; independence. ...

The P=NP problem - New Mexico State University

... • “stochastic” from “stochos”: target, aim, guess • Early Christians: every event, no matter how trivial, was perceived to be a direct manifestation of God’s deliberate intervention • St. Augustine: “We say that those causes that are said to be by chance are not nonexistent but are hidden, and we at ...

... • “stochastic” from “stochos”: target, aim, guess • Early Christians: every event, no matter how trivial, was perceived to be a direct manifestation of God’s deliberate intervention • St. Augustine: “We say that those causes that are said to be by chance are not nonexistent but are hidden, and we at ...

DevStat8e_03_06

... The Poisson Probability Distribution The binomial, hypergeometric, and negative binomial distributions were all derived by starting with an experiment consisting of trials or draws and applying the laws of probability to various outcomes of the experiment. There is no simple experiment on which the ...

... The Poisson Probability Distribution The binomial, hypergeometric, and negative binomial distributions were all derived by starting with an experiment consisting of trials or draws and applying the laws of probability to various outcomes of the experiment. There is no simple experiment on which the ...

The Poisson process Math 217 Probability and Statistics

... time. We’ve used the word “event” to mean one thing so far, namely a subset of a sample space, but we’ll also use it for each occurrence that occurs in a Poisson process. An example of random events over time is radioactive decay. Suppose we have a mass of some radioactive substance. A Geiger counte ...

... time. We’ve used the word “event” to mean one thing so far, namely a subset of a sample space, but we’ll also use it for each occurrence that occurs in a Poisson process. An example of random events over time is radioactive decay. Suppose we have a mass of some radioactive substance. A Geiger counte ...

Bayesian Networks - Blog of Applied Algorithm Lab., KAIST

... • Assign a prior distribution over network structures and find the most likely network by combining its prior probability with the probability accorded to the network by the data. ...

... • Assign a prior distribution over network structures and find the most likely network by combining its prior probability with the probability accorded to the network by the data. ...

Basic Probability

... means that the probability of there being a spike in any small time interval of length t (sec) is t. One may say that Poisson spiking is “as random as it gets.” Remark: Let us chop up the time axis in very small time intervals, say of length 1 sec = 10-6 sec. Consider a Poisson neuron with firin ...

... means that the probability of there being a spike in any small time interval of length t (sec) is t. One may say that Poisson spiking is “as random as it gets.” Remark: Let us chop up the time axis in very small time intervals, say of length 1 sec = 10-6 sec. Consider a Poisson neuron with firin ...

Model neurons

... the instantaneous firing rate. It follows that the generation of each spike is independent of all the other spikes, hence we refer to this as the independent spike hypothesis.! If the independent spike hypothesis were true, then the spike train would be completely described a particular kind of rand ...

... the instantaneous firing rate. It follows that the generation of each spike is independent of all the other spikes, hence we refer to this as the independent spike hypothesis.! If the independent spike hypothesis were true, then the spike train would be completely described a particular kind of rand ...

PowerPoint Link - Personal.psu.edu

... phenomena where the exact outcome is not known but probabilities can be determined. One example is an analysis of population movement between cities and suburbs in the United States. Recall some basic ideas of probability. If the outcome of an event is sure to occur, we say the probability of the ou ...

... phenomena where the exact outcome is not known but probabilities can be determined. One example is an analysis of population movement between cities and suburbs in the United States. Recall some basic ideas of probability. If the outcome of an event is sure to occur, we say the probability of the ou ...

New Advances for the Analysis Tracking of Networks A White Paper

... like to know that the hierarchical expansions are in some sense “complete”. In summary, we would like to have something like the expansion of sound waves in harmonics (Fourier analysis of orthogonal functions). What do people do without well defined metrics? Actually they can make a lot of progress ...

... like to know that the hierarchical expansions are in some sense “complete”. In summary, we would like to have something like the expansion of sound waves in harmonics (Fourier analysis of orthogonal functions). What do people do without well defined metrics? Actually they can make a lot of progress ...

Notes on Special Discrete Distributions

... and var(X) = E(X 2 ) − [E(X)]2 works out to give var(X) = r(1 − p)/p2 . The cdf, F (x), can be obtained from p(x) in the usual way, as a summation, but has no nice closed formula. A NegBin(r, p) random variable X can be thought of as the number of trials until the rth success when independent trials ...

... and var(X) = E(X 2 ) − [E(X)]2 works out to give var(X) = r(1 − p)/p2 . The cdf, F (x), can be obtained from p(x) in the usual way, as a summation, but has no nice closed formula. A NegBin(r, p) random variable X can be thought of as the number of trials until the rth success when independent trials ...

A-01-Introduction

... No. There are none ! 1. No more independent axioms of the form 1 & … & n where each statement stands for an independence statement. 2. We use the symbol for a set of independence statements. In this notation: is derivable from via these properties if and only if is entailed by (i ...

... No. There are none ! 1. No more independent axioms of the form 1 & … & n where each statement stands for an independence statement. 2. We use the symbol for a set of independence statements. In this notation: is derivable from via these properties if and only if is entailed by (i ...

Louigi Addario-Berry Growing - Duke Mathematics Department

... Here, u = u(t, x) ∈ Rd is the position of the polymer, x ∈ R is the length along the polymer, and t is time. Ẇ (t, x) is two-parameter vector-valued white noise. Note that u ∈ Rd ; that is, u is vector-valued. This is consistent with the interpretation of u as the position of a polymer. We say that ...

... Here, u = u(t, x) ∈ Rd is the position of the polymer, x ∈ R is the length along the polymer, and t is time. Ẇ (t, x) is two-parameter vector-valued white noise. Note that u ∈ Rd ; that is, u is vector-valued. This is consistent with the interpretation of u as the position of a polymer. We say that ...

Example Consider tossing a coin 15 times and let X=number of

... The Poisson distribution is an infinite discrete probability distribution that tells you the probability of a certain number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independent of each other. You should think about the Poisson dist ...

... The Poisson distribution is an infinite discrete probability distribution that tells you the probability of a certain number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independent of each other. You should think about the Poisson dist ...

A Review on `Probability and Stochastic Processes`

... such as economics or sociology. But recent research also recognizes that these topics are highly important in natural sciences such as physics, biology or psychology because natural laws are rather probabilistic than deterministic. Of course, one reason why the book is not recognized by a broad audi ...

... such as economics or sociology. But recent research also recognizes that these topics are highly important in natural sciences such as physics, biology or psychology because natural laws are rather probabilistic than deterministic. Of course, one reason why the book is not recognized by a broad audi ...

Workshop on Stochastic Differential Equations and

... Chapman Kolmogorov Equations • For a continuous time continuous space process the Chapman Kolmogorov Equations are • If • The C-K equation in this case become ...

... Chapman Kolmogorov Equations • For a continuous time continuous space process the Chapman Kolmogorov Equations are • If • The C-K equation in this case become ...

More on the exponential- distribution and Poisson processes.

... is not exponentially distributed!) Further we have learnt that a sum of independent identically exponentially distributed variables is having a gamma distribution. If X1 , . . . , Xk are indep. exponentially distributed k with expectation β then Y = i=1 is gamma distributed with parameters α = k an ...

... is not exponentially distributed!) Further we have learnt that a sum of independent identically exponentially distributed variables is having a gamma distribution. If X1 , . . . , Xk are indep. exponentially distributed k with expectation β then Y = i=1 is gamma distributed with parameters α = k an ...

On the intersections between the trajectories of a

... Accordingly it m a y be supposed that, subject to appropriate conditions on ~(t), the stream of upcrossings will tend to form a Poisson process as the level u tends to infinity. T h a t this is actually so was first proved b y Volkonskij and Rozanov in their remarkable joint paper [8]. They assumed, ...

... Accordingly it m a y be supposed that, subject to appropriate conditions on ~(t), the stream of upcrossings will tend to form a Poisson process as the level u tends to infinity. T h a t this is actually so was first proved b y Volkonskij and Rozanov in their remarkable joint paper [8]. They assumed, ...

The Poisson Process

... Let us assume that we are counting the occurrence of rare events over a period of time (or distance, or volume, etc.). We will assume that for some constant > 0, the following statements hold true: 1) The probability that exactly one event occurs in an interval of length h is the same for all such ...

... Let us assume that we are counting the occurrence of rare events over a period of time (or distance, or volume, etc.). We will assume that for some constant > 0, the following statements hold true: 1) The probability that exactly one event occurs in an interval of length h is the same for all such ...

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.