Discrete probability spaces

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  • The origins of this book lie in our earlier book Random Processes: A Mathematical Approach for Engineers, Prentice Hall, 1986. This book began as a second edition to the earlier book and the basic goal remains unchanged | to introduce the fundamental ideas and mechanics of random processes to engineers in a way that accurately reects the underlying mathematics, but does not require an extensive mathematical background and does not belabor detailed general proofs when simple cases suce to get the basic ideas across....

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  • In this chapter, we study the mathematical structure of a simple one-period model of a financial market. We consider a finite number of assets. Their initial prices at time t = 0 are known, their future prices at time t = 1 are described as random variables on some probability space. Trading takes place at time t = 0. Already in this simple model, some basic principles of mathematical finance appear very clearly. In Section 1.2, we single out those models which satisfy a condition of market efficiency: There are no trading opportunities which yield a profit without any downside risk.

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  • Chapter 4 Conditional Probability 4.1 Discrete Conditional Probability In this section we ask and answer the following question. Suppose we assign a distribution function to a sample space and then learn that an event E has occurred. How should we change the probabilities of the remaining events?

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  • Chapter 12 Random Walks 12.1 Random Walks in Euclidean Space In the last several chapters, we have studied sums of random variables with the goal being to describe the distribution and density functions of the sum. In this chapter, we shall look at sums of discrete random variables from a different perspective.

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