Solving Quadratic Equations via PhaseLift when There Are About As
... where c0 is a sufficiently large constant. Thus, exact recovery holds simultaneously over all input signals. In words, (1) the solution to most systems of quadratic equations can be obtained by semidefinite programming as long as the number of equations is at least a constant times the number of unk ...
... where c0 is a sufficiently large constant. Thus, exact recovery holds simultaneously over all input signals. In words, (1) the solution to most systems of quadratic equations can be obtained by semidefinite programming as long as the number of equations is at least a constant times the number of unk ...
1 Gambler`s Ruin Problem
... Recurrence of the simple symmetric random walk Combining the results for both M and m in the previous section when p = 0.5, we have Proposition 1.4 The simple symmetric (p = 0.50) random walk, starting at the origin, will wp1 eventually hit any integer a, positive or negative. In fact it will hit an ...
... Recurrence of the simple symmetric random walk Combining the results for both M and m in the previous section when p = 0.5, we have Proposition 1.4 The simple symmetric (p = 0.50) random walk, starting at the origin, will wp1 eventually hit any integer a, positive or negative. In fact it will hit an ...
Parameter Estimation
... Often an estimator’s standard error is a function of θ or other parameters; these must be replaced by estimates before we can actually calculate a value. Estimated standard error The resulting statistic is called the estimated standard error, and is denoted σ̂θ̂ . Example: binomial distribution; V ( ...
... Often an estimator’s standard error is a function of θ or other parameters; these must be replaced by estimates before we can actually calculate a value. Estimated standard error The resulting statistic is called the estimated standard error, and is denoted σ̂θ̂ . Example: binomial distribution; V ( ...
Sampling Theory VARYING PROBABILITY SAMPLING
... If y is the variable under study and x is an auxiliary variable related to y, then in the most commonly used varying probability scheme, the units are selected with probability proportional to the value of x, called as size. This is termed as probability proportional to a given measure of size (pps) ...
... If y is the variable under study and x is an auxiliary variable related to y, then in the most commonly used varying probability scheme, the units are selected with probability proportional to the value of x, called as size. This is termed as probability proportional to a given measure of size (pps) ...
3 Probabilistic Turing Machines
... When we talked about computability, we found that nondeterminism was sometimes a convenient mechanism for designing algorithms but that it could be simulated deterministically; that is, we found that “exotic” nondeterministic algorithms have no greater computational ability than do their mundane det ...
... When we talked about computability, we found that nondeterminism was sometimes a convenient mechanism for designing algorithms but that it could be simulated deterministically; that is, we found that “exotic” nondeterministic algorithms have no greater computational ability than do their mundane det ...
Keywords Limiting probability, Probability of state, Markov Processes
... In order to find unconditional probabilities we need to know initial probability distribution, i.e. probabilities at the beginning of the process at time t0 = 0 : p1 (0) , p2 (0) ,…, ...
... In order to find unconditional probabilities we need to know initial probability distribution, i.e. probabilities at the beginning of the process at time t0 = 0 : p1 (0) , p2 (0) ,…, ...
I.I.D. Random Variables - inst.eecs.berkeley.edu
... Before we prove this theorem, let us formalize something we have been assuming implicitly for some time: Joint Distributions Consider two random variables X and Y defined on the same probability space. By linearity of expectation, we know that E(X +Y ) = E(X)+E(Y ). Since E(X) can be calculated if w ...
... Before we prove this theorem, let us formalize something we have been assuming implicitly for some time: Joint Distributions Consider two random variables X and Y defined on the same probability space. By linearity of expectation, we know that E(X +Y ) = E(X)+E(Y ). Since E(X) can be calculated if w ...
The Annals of Mathematical Statistics, 38, 1967, pp. 780-786.
... measure have been presented by de Finetti [4], Koopman [5], [B], [7], and Savage [ll]. They all involve, either explicitly or implicitly, the strong property that for each integer n the universal event can be partitioned into n events that are equally probable under the given ordering. Among other t ...
... measure have been presented by de Finetti [4], Koopman [5], [B], [7], and Savage [ll]. They all involve, either explicitly or implicitly, the strong property that for each integer n the universal event can be partitioned into n events that are equally probable under the given ordering. Among other t ...
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... In casinos, there is often a large display next to every roulette table showing the outcomes of the last several spins of the wheel. Since the results of previous spins reveal nothing about the results of future spins, why do the casinos pay for these displays? ...
... In casinos, there is often a large display next to every roulette table showing the outcomes of the last several spins of the wheel. Since the results of previous spins reveal nothing about the results of future spins, why do the casinos pay for these displays? ...
Introduction to Probability I
... Bayesian (subjective) probability - reflects a person’s opinion about how likely an event is to occur, it represents the person’s strength of belief – Prior beliefs are updated using observed data – Valid in situations where long run relative frequency is not applicable ...
... Bayesian (subjective) probability - reflects a person’s opinion about how likely an event is to occur, it represents the person’s strength of belief – Prior beliefs are updated using observed data – Valid in situations where long run relative frequency is not applicable ...
Moment generating functions
... with parameter λp. Using the inversion theorem, we conclude that Y is exponentially distributed. In view of the fact that the sum of a fixed number of exponential random variables is far from exponential, this result is rather surprising. An intuitive explanation will be provided later in terms of th ...
... with parameter λp. Using the inversion theorem, we conclude that Y is exponentially distributed. In view of the fact that the sum of a fixed number of exponential random variables is far from exponential, this result is rather surprising. An intuitive explanation will be provided later in terms of th ...
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... in Los Angeles. If one person is selected from each city, what is the probability that all of them are under 18? Is this an unusual event? Solution: These three events are independent because the identity of the person chosen from one city does not affect who is chosen in the other cities. Therefore ...
... in Los Angeles. If one person is selected from each city, what is the probability that all of them are under 18? Is this an unusual event? Solution: These three events are independent because the identity of the person chosen from one city does not affect who is chosen in the other cities. Therefore ...
Determine whether the events are independent or dependent. Then
... 2. Yana has 4 black socks, 6 blue socks, and 8 white socks in his drawer. If he selects three socks at random with no replacement, what is the probability that he will first select a blue sock, then a black sock, and then another blue sock? SOLUTION: Since the socks are being selected with out rep ...
... 2. Yana has 4 black socks, 6 blue socks, and 8 white socks in his drawer. If he selects three socks at random with no replacement, what is the probability that he will first select a blue sock, then a black sock, and then another blue sock? SOLUTION: Since the socks are being selected with out rep ...
Probability box
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A probability box (or p-box) is a characterization of an uncertain number consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed. Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes.An example p-box is shown in the figure at right for an uncertain number x consisting of a left (upper) bound and a right (lower) bound on the probability distribution for x. The bounds are coincident for values of x below 0 and above 24. The bounds may have almost any shapes, including step functions, so long as they are monotonically increasing and do not cross each other. A p-box is used to express simultaneously incertitude (epistemic uncertainty), which is represented by the breadth between the left and right edges of the p-box, and variability (aleatory uncertainty), which is represented by the overall slant of the p-box.