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 ...
... 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
... 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 ...
... 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 ...
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 ...
... 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
... ◮ 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 ...
... ◮ 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 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− ...
... 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
... [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 ...
... [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 ...
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, ... ...
... 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, ... ...
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 ...
... 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 ...
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) = ...
... 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
... 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 ...
... 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 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 ...
... 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 ...