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The Idea of Probability
The Idea of Probability

What Conditional Probability Must (Almost) Be
What Conditional Probability Must (Almost) Be

Using Metrics in Stability of Stochastic Programming Problems
Using Metrics in Stability of Stochastic Programming Problems

Lecture 1: simple random walk in 1-d Today let`s talk about ordinary
Lecture 1: simple random walk in 1-d Today let`s talk about ordinary

... Lecture 1: simple random walk in 1-d Today let’s talk about ordinary simple random walk to introduce ourselves to some of the questions. Starting in d = 1, Definition 0.1. Let Y1 , . . . be a sequence of i.i.d. random variables defined on some probability space (Ω, F, P) with P(Y1 = 1) = 1/2 = P(Y1 ...
BROWNIAN MOTION Contents 1. Continuous Random Variables 1
BROWNIAN MOTION Contents 1. Continuous Random Variables 1

... Since this paper deals primarily with a stochastic process, a sequence of random variables indexed by time, we are first going to need to know a little bit of the machinery of probability in order to achieve any useful results. We assume that the reader has some familiarity with basic (discrete) pro ...
A Note on Coloring Random k-Sets
A Note on Coloring Random k-Sets

Lecture 10: Hard-core predicates 1 The Next
Lecture 10: Hard-core predicates 1 The Next

... Definition 1 (Next-bit test) An ensemble of probability distributions {Xn } over {0, 1}m(n) is said to pass the next bit test if ∃ a negligible function ǫ(n) so that ∀ nonuniform PPT A and ∀n ∈ N it holds that P r[t ← Xn A(t0→i ) = ti+1 ] ≤ 12 + ǫ(n) Note that the uniform distribution passes the nex ...
Machine Learning: Probability Theory
Machine Learning: Probability Theory

... ◮ F is monotonically increasing, limx→−∞ F (x) = 0, limx→∞ F (x) = 1 ◮ if exists, the derivative of F is called a probability density function (pdf). It yields large values in the areas of large probability and small values in the areas with small probability. But: the value of a pdf cannot be inter ...
Chapter 7 Probability Distributions, Information about the Future
Chapter 7 Probability Distributions, Information about the Future

Big Outliers Versus Heavy Tails: what to use?
Big Outliers Versus Heavy Tails: what to use?

Chapter 4 Dependent Random Variables
Chapter 4 Dependent Random Variables

... exist sets An with λ(A) ≥ m+ − n1 and therefore totally positive subsets Ān of An with λ(Ān ) ≥ m+ − n1 . Clearly Ω+ = ∪n Ān is totally positive and λ(Ω+ ) = m+ . It is easy to see that Ω− = Ω − Ω+ is totally negative. µ± can be taken to be the restriction of λ to Ω± . Remark 4.2. If λ = µ+ − µ− ...
Exact upper tail probabilities of random series
Exact upper tail probabilities of random series

... [15], [7] and [12], several estimates were obtained on the upper tail probabilities for suitable random variables, but those estimates are not exact. The first exact upper tail probability was derived in [19] with i.i.d. nonnegative {ξj } having regular variation at infinity, where the coefficients ...
Members of random closed sets - University of Hawaii Mathematics
Members of random closed sets - University of Hawaii Mathematics

Lecutre 19: Witness-Hiding Protocols and MACs (Nov 3, Gabriel Bender)
Lecutre 19: Witness-Hiding Protocols and MACs (Nov 3, Gabriel Bender)

No Slide Title - Lyle School of Engineering
No Slide Title - Lyle School of Engineering

... X = number of failures that precede the rth success X is called the negative binomial random variable because, in contrast to the binomial random variable, the number of successes is fixed and the number of trials is random. Possible values of X are x = 0, 1, 2, ... ...
Probability Distribution
Probability Distribution

Lecture 2: Random variables in Banach spaces
Lecture 2: Random variables in Banach spaces

Notes from Week 9: Multi-Armed Bandit Problems II 1 Information
Notes from Week 9: Multi-Armed Bandit Problems II 1 Information

... Proof. The intuition is as follows. Let Qj denote the random variable which counts the number of times ALG flips coin j. If Ej (Qj ) is much smaller than 1/ε2 , then at time t the algorithm is unlikely to have accumulated enough evidence that j is the biased coin. On the other hand, since there are ...
Precalculus Module 5, Topic B, Lesson 10: Student
Precalculus Module 5, Topic B, Lesson 10: Student

Markov and Chebyshev`s Inequalities
Markov and Chebyshev`s Inequalities

... Question: A biased coin is flipped 200 times consecutively, and comes up heads with probability 1/10 each time it is flipped. Give an upper bound the probability that it will come up heads at least 120 times. Solution: Let X be the r.v. that counts the number of heads. Recall: E(X ) = 200 ∗ (1/10) = ...
Zeros of Gaussian analytic functions—invariance and rigidity
Zeros of Gaussian analytic functions—invariance and rigidity

... analytic function f(z) = ξk / k!zk . The resulting process is translation invariant and ergodic. Theorem 1. (Sodin rigidity) f(z) is the unique Gaussian entire function with a translation invariant zero process of intensity 1. A Gaussian analytic function is one for which (f(z1 ), . . . f(zk )) is a ...
Lecture Notes on Statistical Methods
Lecture Notes on Statistical Methods

Lecture 6: State-Based Methods (cont 2)
Lecture 6: State-Based Methods (cont 2)

... 2 (takes on a value of 2) with an exponentially distributed time with parameter  . Independently, X goes to state 3 with an exponentially distributed time with parameter . These state transitions are like competing random variables. We say that from state 1, X goes to state 2 with rate  and to s ...
Lecture 11 1 Recap 2 Amplification for BPP 3 BPP ⊆ P/poly
Lecture 11 1 Recap 2 Amplification for BPP 3 BPP ⊆ P/poly

binomial_old
binomial_old

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Conditioning (probability)

Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations and conditional distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random result if the condition is completely specified; otherwise, if the condition is left random, the result of conditioning is also random.This article concentrates on interrelations between various kinds of conditioning, shown mostly by examples. For systematic treatment (and corresponding literature) see more specialized articles mentioned below.
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