File Name: probability theory and stochastic processes by srihari .zip
About this page. In a simple, generic form we can write this process as x p x jy The data-generating distribution.
- PROBABILITY THEORY AND STOCHASTIC PROCESS SRIHARI PDF
- PROBABILITY THEORY AND STOCHASTIC PROCESSES BY P SRIHARI PDF
- Probability Theory and Stochastic Processes
- Course Description
Unequal Distribution, Equal Distributions. Spectral characteistics of system response: Power density spectrum of response, cross power spectral density of input and output of a linear system. Peebles, TMH, 4th Edition, Singh and S.
PROBABILITY THEORY AND STOCHASTIC PROCESS SRIHARI PDF
Sign in. The answer to the above question is the main motivation behind this article. The study of these quantities is quite different from deterministic quantities arising in a range of computer science fields.
Given this crucial information, it is therefore desirable to be able to reason in an environment of uncertainty, and probability theory is the tool that shall help us to do so. Because I do not want to cloud your thoughts with mathematical jargon in the very beginning, I have added a section on application of all these things at the very end of the article.
T hat should be your prime motivation for understanding this stuff. Let us es t ablish a bit of mathematical literature in terms of which we can base our further discussion.
Firstly, a deterministic system can be thought of as something which involves absolutely no randomness in the development of future states. One can determine the future state of an accelerating system. The outcome is not random. On the other hand, a non-deterministic system is something that is not deterministic or involves a fair amount of randomness in future states. For instance, flipping a coin is a non-deterministic process as there is randomness involved in the result either head or tails and there is no way to ascertain which outcome shall come.
Back to the question. There can be a variety of ways due to which stochastic behaviour is introduced in systems. A system might be inherently stochastic , like a Quantum Mechanics model.
In such a case, there is no way we can make deterministic arguments about the state of the system. Or there might be a system that is deterministic given we have complete knowledge about the variables of the system.
Now if we lose some knowledge about these variables, we lose the ability to be able to determine the future states of development of the system. And thus, a deterministic system turns into a non-deterministic one. Reading this article requires a basic understanding of the notion of probability, some idea about frequentist and Bayesian probability, basic idea about conditional probability, notions on in dependent events.
As detailed above, a non-deterministic system may have more than one possible outcomes. For instance, flipping a coin may have two different, equally likely outcomes- heads or tails. A random variable or Stochastic variable can be thought of as a variable whose possible values are the outcomes of the non-deterministic system being modelled. For instance, we can define a random variable X that shall denote the outcome of a coin flip.
A random variable may be discrete if it covers finite or countably infinite number of states or continuous if it covers uncountably infinite number of states.
Note : The notion of countably infinite and uncountably infinite merits a whole article explaining it and is thus omitted here. You can, however, view ideas on set domination online. I am attaching a very brief discussion here. Consider two sets- X and N the set of Natural numbers and the usual definitions of mapping and bijection.
Set X is said to strictly dominate N if there exists a mapping from N to a subset of X and not to the whole of X. In other words, there exists at least one element in X which has no pre-image in N.
You can construct similar condition for the set N strictly dominating the set X. Also, the set X is said to be equivalent to the set N when there exists a bijection between the two.
Now, X is finite when N strictly dominates X. X is countably infinite when X is equivalent to N. And X is uncountably infinite when X strictly dominates N. Simply stating, PDF tells you how likely is a random variable to take on a particular value.
Formally stating,. This is the probability distribution function of a discrete random variable. Consider the experiment of throwing two dice and let X be a random variable depicting the sum of the numbers of individual dices. You can see here how the values states of X are mapped to their respective probabilities in the table defined above.
You may find more information about how to go about calculating this here. This is the probability distribution function of a continuous variable. The probability associated can be given by f x. You may also want to read about bivariate distribution functions joint probability distribution functions discrete and continuous and marginal distribution functions. Expectation value of a random variable can be thought of as the mean value the variable takes when it is drawn according to the probability distribution f x.
Calculations are done as follows:. Likewise, variance of a random variable can be seen as a measure of how much the values of a function of a random variable vary when X is drawn from a probability distribution f x.
Variance is the expectation value of the square of X — mean. A very detailed theory and practice on expectation values can be found here. Covariance is a sense of how much variables are related to each other. Take for instance this covariance matrix:. In this matrix in the first row, is the variance of A, is the covariance of A and B, is the covariance of A and C and so on. Figure 5 shows the calculation of the covariances depicted in the table above, where f x, y is the joint probability distribution of random variables X and Y.
From the table, certain deductions can be drawn. Vice-versa case for positive covariance both variables tend to take either high or low values simultaneously. There are several predefined probability mass and probability density functions. This is the distribution function over a binary, single, discrete random variable or a discrete random variable X that can take only 2 values.
Formally, the Bernoulli distribution is parameterised by a single parameter denoting the probability of success or whose value is equal to p if you consider the example in the last paragraph :. Now consider the following:. Here, the probability that X takes the value 1 or head is tossed as in our example is given by the parameter phi which takes some value between 0 and 1. Likewise, the chance of the other event happening tails being tossed is 1 — phi.
We can combine both these probabilities into a generalised statement given by:. Utilising the concept of expectation values and variance as discussed above, one can find the mean and variance of this distribution as:.
The informal meaning of sum of n independent and identically distributed Bernoulli variables is that we repeat the same experiment n times, and the outcome of each experiment is independent of the outcome of the others. We also define a parameter p which is identical to the parameter phi in Bernoulli distribution that denotes the probability of the random variable taking on the value 1 in that instance of the experiment out of the n instances of the experiment.
The binomial distribution thus goes like:. For instance, take 5 tosses of a fair and balanced coin. Now define a random variable X that denotes the number of heads obtained. Formally stating, if we define the Bernoulli variable X[i] as the outcome of the i th coin toss, we need to add X, X, … , X in order to get our desired value of X. This is the most basic distribution function for continuous random variables.
This is parameterised by the mean and variance denoted by their standard symbols of the distribution as follows:. The function is graphed as follows:. It is therefore assumed that values will follow a normal distribution with an equal number of measurements above and below the mean value, with the mean value being the peak of the distribution.
It would be sensible, in this case, to make the least number of assumptions about the distribution of the variable and choose Gaussian distribution function which introduces the maximum amount of uncertainty over the distribution of X among all the distributions with finite variance.
Condensing the above paragraph into a statement, it is highly probable that your continuous random variable X follows a Gaussian distribution with some noise suggested by the Central Limit theorem. So why not make that assumption beforehand? In case we wish to model multivariate distributions, we can have the Gaussian distribution as:. Find more about Gaussian distributions in multivariate settings in Gaussian Mixture model discussed below.
In Deep Learning, we need to regularise the parameters of a neural network to prevent overfitting. From a Bayesian perspective, fitting a regularised model can be interpreted as computing the maximum a posteriori MAP estimate.
We hence use the following:. The indicator function serves to assign a 0 probability to all negative values of X. You can view the graph:. The exponential distribution describes the time between the events in a Poisson point process, i. Similarly, if we wish to model a spike at the mean of the distribution of X, we can use the Laplace distribution function,. This Laplacian distribution can be viewed as two exponential distributions spliced together back-to-back such that we obtain a spike at the mean of the distribution note how we get a shift in the green curve in the above figure, this can be managed by the parameter mean in the Laplacian distribution.
The Dirac Delta function serves to cluster all the distribution around a single point and neutralises distributions over all other values of the continuous random variable X. The above equation tends to collect all the mass around the mean. A more specific example is the following function:.
Here, a is a parameter whose value serves to define the closeness of the peak to the origin or the concentration of the mass about the origin. As a approaches 0, the peak becomes infinitely narrow and infinitely high.
You can parameterise the equation in Figure 19 with the mean. The function would then look like the equation in figure 18 and would result in a peak at the desired spot other than 0. The Dirac Delta functions find their importance in the next distribution:. Empirical distribution is the multinoulli distribution of continuous random variables.
Consider the following:. Find more explanation on this in the next section…. Congratulations if you have made this far!
PROBABILITY THEORY AND STOCHASTIC PROCESSES BY P SRIHARI PDF
Many familiar dilemmas that we find in the application of data-driven AI have their origins in technical-mathematical choices that we have made along the way to this version of AI. Several of them might need to be reconsidered in order for the field to move forward. After reviewing some of the current problems related to AI, we trace their cultural, technical and economic origins, then we discuss possible solutions. His research covers machine learning methods, and applications of AI to the analysis of media content, as well as the social and ethical implications of AI. Cristianini is the co-author of two widely known books in machine learning, as well as a book in bioinformatics.
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Probability Theory and Stochastic Processes
Sign in. The answer to the above question is the main motivation behind this article. The study of these quantities is quite different from deterministic quantities arising in a range of computer science fields. Given this crucial information, it is therefore desirable to be able to reason in an environment of uncertainty, and probability theory is the tool that shall help us to do so.
If three of these companies are chosen at random without replacement, what is the probability that each of the three has installed WLANS. A clour blind person is chosen at random. What is the probability that the person is a male? Also evaluate P X0.
Ему предложили исчезнуть. - Диагностика, черт меня дери! - бормотал Чатрукьян, направляясь в свою лабораторию. - Что же это за цикличная функция, над которой три миллиона процессоров бьются уже шестнадцать часов. Он постоял в нерешительности, раздумывая, не следует ли поставить в известность начальника лаборатории безопасности. Да будь они прокляты, эти криптографы. Ничего не понимают в системах безопасности.
Doob's decomposition of a stochastic process Doob's These notes grew from an introduction to probability theory taught during the first and second with the probability density function (PDF) fx(s) = F^(s) for continuous random.
Probability Density Function
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