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...dom matrix <math>A\in\mathbf{R}^{k\times d}</math> and show that with high probability (<math>1-O(1/n)</math>) it is a good embedding satisfying: and the probability density function is given by <math>p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\mathr ...

...dom matrix <math>A\in\mathbf{R}^{k\times d}</math> and show that with high probability (<math>1-O(1/n)</math>) it is a good embedding satisfying: and the probability density function is given by <math>p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\mathr ...

...dom matrix <math>A\in\mathbf{R}^{k\times d}</math> and show that with high probability (<math>1-O(1/n)</math>) it is a good embedding satisfying: and the probability density function is given by <math>p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\mathr ...

...dom matrix <math>A\in\mathbf{R}^{k\times d}</math> and show that with high probability (<math>1-O(1/n)</math>) it is a good embedding satisfying: and the probability density function is given by <math>p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\mathr ...

* A matrix whose entries follow i.i.d. normal distributions. (Due to Indyk-Motwani) ...random projection <math>A</math> violates the distortion requirement with probability at most ...

* A matrix whose entries follow i.i.d. normal distributions. (Due to Indyk-Motwani) ...random projection <math>A</math> violates the distortion requirement with probability at most ...

...ath>" for the event <math>\{a\in\Omega\mid X(a)=x\}</math>, and denote the probability of the event by Note that in probability theory, the "mutual independence" is <font color="red">not</font> equivalen ...

...nd <math>\Psi_X^*(t)</math>? And give a tail inequality to upper bound the probability <math>\Pr[X\ge t]</math>. ...nd <math>\Psi_X^*(t)</math>? And give a tail inequality to upper bound the probability <math>\Pr[X\ge t]</math>. ...

The <i>Chernoff bound</i> is an exponentially decreasing bound on tail distributions. Let <math>X_1,\dots,X_n</math> be independent random variables and <math>\ ...integer programming and puts vertex <math>i</math> in <math>S</math> with probability <math>f(x_i^*)</math>. We may assume that <math>f(x)</math> is a linear fun ...

== Probability Basics == In probability theory class we have learned the basic concepts of '''events''' and '''rand ...

To see this, we apply the law of total probability, ...tep, at the current node, the walk moves through an adjacent edge with the probability of the weight of the edge. It is easy to see that this is a well-defined ra ...

...ath>" for the event <math>\{a\in\Omega\mid X(a)=x\}</math>, and denote the probability of the event by Note that in probability theory, the "mutual independence" is <font color="red">not</font> equivalen ...

=Probability Space= The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogoro ...

To see this, we apply the law of total probability, ...tep, at the current node, the walk moves through an adjacent edge with the probability of the weight of the edge. It is easy to see that this is a well-defined ra ...

* birthday problem: the probability that every bin contains at most one ball (the mapping is 1-1); * coupon collector problem: the probability that every bin contains at least one ball (the mapping is on-to); ...

* birthday problem: the probability that every bin contains at most one ball (the mapping is 1-1); * coupon collector problem: the probability that every bin contains at least one ball (the mapping is on-to); ...

* birthday problem: the probability that every bin contains at most one ball (the mapping is 1-1); * coupon collector problem: the probability that every bin contains at least one ball (the mapping is on-to); ...

* birthday problem: the probability that every bin contains at most one ball (the mapping is 1-1); * coupon collector problem: the probability that every bin contains at least one ball (the mapping is on-to); ...

* birthday problem: the probability that every bin contains at most one ball (the mapping is 1-1); * coupon collector problem: the probability that every bin contains at least one ball (the mapping is on-to); ...

...ices. We will show that <math>G(V,E)</math> is an expander graph with high probability. Formally, for some constant <math>d</math> and constant <math>\alpha</math ...ic method, this shows that there exist expander graphs. In fact, the above probability bound shows something much stronger: it shows that almost every regular gra ...