In this document we discuss the meaning of conditioning on certain
... target space of Y may not be Polish. But usually we will need some niceness of the target space because often times these strong regularity hypotheses can only be checked when everything is nice and Euclidean-smooth, at least in finite dimensional cross-sections. Even so, situations where one is try ...
... target space of Y may not be Polish. But usually we will need some niceness of the target space because often times these strong regularity hypotheses can only be checked when everything is nice and Euclidean-smooth, at least in finite dimensional cross-sections. Even so, situations where one is try ...
6.436J Lecture 17: Convergence of random
... Xn converges to Y in distribution. However, for almost all ω, the sequence Xn (ω) does not converge. (e) If we are dealing with random variables whose distribution is in a parametric class, (e.g., if every Xn is exponential with parameter λn ), and the parame ters converge (e.g., if λn → λ > 0 and ...
... Xn converges to Y in distribution. However, for almost all ω, the sequence Xn (ω) does not converge. (e) If we are dealing with random variables whose distribution is in a parametric class, (e.g., if every Xn is exponential with parameter λn ), and the parame ters converge (e.g., if λn → λ > 0 and ...
Solving Quadratic Equations via PhaseLift when There Are About As
... where β0 = E z 4 1(|z| ≤ 3) ≈ 2.6728 with z ∼ N (0, 1), is a valid certificate. As claimed, kλk∞ ≤ 7/m. We begin by checking the condition YT ⊥ −IT ⊥ . First, the matrix Y (1) is Wishart and standard results in random matrix theory—e.g. Corollary 5.35 in [5]—assert that ...
... where β0 = E z 4 1(|z| ≤ 3) ≈ 2.6728 with z ∼ N (0, 1), is a valid certificate. As claimed, kλk∞ ≤ 7/m. We begin by checking the condition YT ⊥ −IT ⊥ . First, the matrix Y (1) is Wishart and standard results in random matrix theory—e.g. Corollary 5.35 in [5]—assert that ...
Lecture - Sybil Nelson
... a randomly chosen point on the surface. Let M = the maximum depth (in meters), so that any number in the interval [0, M] is a possible value of X. If we “discretize” X by measuring depth to the nearest meter, then possible values are nonnegative integers less than or equal to M. The resulting discre ...
... a randomly chosen point on the surface. Let M = the maximum depth (in meters), so that any number in the interval [0, M] is a possible value of X. If we “discretize” X by measuring depth to the nearest meter, then possible values are nonnegative integers less than or equal to M. The resulting discre ...
Activity: Determining if a Die is Fair
... up atoms 25% of the time ( f10 5 0.25 ) in this case. The most obvious conclusion is that one single measurement of 10 atoms is not too reliable a predictor of the probability P that an atom is measured to have spin up. To reliably predict the probability we must perform repeated experiments and ...
... up atoms 25% of the time ( f10 5 0.25 ) in this case. The most obvious conclusion is that one single measurement of 10 atoms is not too reliable a predictor of the probability P that an atom is measured to have spin up. To reliably predict the probability we must perform repeated experiments and ...
ECE-316 Tutorial for the week of June 1-5
... A satellite system consists of n components and functions on any given day if at least k of the n components function on that day. On a rainy day each of the components independently functions with probability , whereas on a dry day, they each independently function with probability . If the probabi ...
... A satellite system consists of n components and functions on any given day if at least k of the n components function on that day. On a rainy day each of the components independently functions with probability , whereas on a dry day, they each independently function with probability . If the probabi ...
Document
... if there exists a function p(x,y) such that: p(x,y) = P[X=x and Y=y] Random variables X and Y are jointly continuous if there exists a non-negative function f(x,y) called the joint probability density function of X and Y, such that for all sets of real numbers A and B ...
... if there exists a function p(x,y) such that: p(x,y) = P[X=x and Y=y] Random variables X and Y are jointly continuous if there exists a non-negative function f(x,y) called the joint probability density function of X and Y, such that for all sets of real numbers A and B ...
Creating Probability Models for Simple Events
... looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cas ...
... looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cas ...
Events A1,...An are said to be mutually independent if for all subsets
... 6’ + (1/8) · 6’1” = 6’. But the average gives only limited information about a distribution. Suppose there were instead only people with heights 5’ and 7’, and an equal number of each, then the average would still be 6’ though these are very different distributions. It is useful to characterize the ...
... 6’ + (1/8) · 6’1” = 6’. But the average gives only limited information about a distribution. Suppose there were instead only people with heights 5’ and 7’, and an equal number of each, then the average would still be 6’ though these are very different distributions. It is useful to characterize the ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.