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...describe the Pareto distribution. [[Insurance]] companies often use Pareto distributions to model damage. [[Category:Probability distributions]] ...

{{Probability distribution| pdf_image =[[File:Normal distribution pdf.png|325px|Probability density function for the normal distribution]]<br /><small>The green line i ...

[[Image:Boxplot vs PDF.svg|thumb|350px|[[Boxplot]] and probability density function of a [[normal distribution]] {{nowrap|''N''(0,&thinsp;''σ' ...obability density function in the interval <math>[a,b]</math> yields the [[probability]] that a given [[random variable]] with the given density is contained in t ...

...otal variation distance''' measures the difference between two probability distributions. :Let <math>p</math> and <math>q</math> be two probability distributions over the same finite state space <math>\Omega</math>, the '''total variatio ...

[[File:Gumbel-Density.svg|thumb|260px|Gumbel probability distribution function (PDF)]] The '''Gumbel distribution''' is a [[probability distribution]] of extreme values. ...

...e value of [[Pi (mathematics)|π]]. After placing 30,000 random points, the probability that the estimate for π is within 0.07% of the actual value.]] ...get a result. The algorithm terminates with an answer that is correct with probability <math>p<1</math>. ...

=Coupling of Two Distributions= Coupling is a powerful proof technique in probability theory. It allows us to compare two unrelated variables (or processes) by f ...

'''Probability Theory''' <br> & '''Mathematical Statistics'''</font> |data15 = '''Probability and Random Processes''' (4E) <br> Geoffrey Grimmett and David Stirzaker <br ...

We introduce several important distributions induced by independent coin flips (independent probabilistic experiments), ...single (biased) coin flip. Suppose that we flip a (biased) coin where the probability of HEADS is <math>p</math>. Let <math>X</math> be the 0-1 random variable w ...

Let <math>p</math> and <math>q</math> be two probability distributions over <math>\Omega</math>. Give an explicit construction of a coupling <math # with probability <math>p</math>, do nothing; ...

'''Probability Theory''' <br> & '''Mathematical Statistics'''</font> |data15 = '''Probability and Random Processes''' (4E) <br> Geoffrey Grimmett and David Stirzaker <br ...

{{Probability distribution | '''Student's t-distribution''' is a [[probability distribution]] which was developed by [[William Sealy Gosset]] in 1908. ''S ...

...y define this notion, we need a way of measuring the closeness between two distributions. ...otal variation distance''' measures the difference between two probability distributions. ...

|caption = ''Probability and Computing: Randomized Algorithms and Probabilistic Analysis'', Mitzenma # Probability Basics | [http://tcs.nju.edu.cn/slides/random2011/random2.pdf slides] ...

...n to which the ball is assigned is uniformly and independently chosen, the distributions of the loads of bins are identical. Thus <math>\mathbf{E}[X_i]</math> is th ...mum load is <math>O\left(\frac{\log n}{\log\log n}\right)</math> with high probability. ...

...y define this notion, we need a way of measuring the closeness between two distributions. ...otal variation distance''' measures the difference between two probability distributions. ...

...y define this notion, we need a way of measuring the closeness between two distributions. ...otal variation distance''' measures the difference between two probability distributions. ...

* 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); ...

...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 ...

...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 ...

<p>Without further notice, we are working on probability space <math>(\Omega,\mathcal{F},\mathbf{Pr})</math>.</p> ...th>0</math>; that is, <math>X_r</math> and <math>-X_r</math> have the same distributions. Show that, for all <math>x</math>, <math>\mathbf{Pr}[S_n \geq x] = \mathbf ...

:The choice of the error probability is arbitrary. In fact, replacing the 1/2 with any constant <math>0<p<1</mat :Replacing the error probability from <math>\frac{1}{4}</math> to <math>\frac{1}{3}</math>, or any constant ...

...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 ...

* 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 ...

...the mixing time, we need a notion of the distance between two probability distributions. Let <math>p</math> and <math>q</math> be two probability distributions over the same finite state space <math>\mathcal{S}</math>, the '''total var ...

-1 & \mbox{with probability }\frac{1}{2}\\ +1 &\mbox{with probability }\frac{1}{2} ...

...ows to find common [[pattern]]s. In statistics, such patterns are called [[probability distribution]]s. The basic idea is to look at the results of an experiment ...refore make a [[Mathematical model|model]] that says that there's a high [[probability]], the [[offspring]] will be small in size if the parents were small in siz ...

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 ...

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 ...

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 ...

...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 ...

:* We choose a country at random with a probability ''proportional to its population'', and <math>\mathbf{E}[Y\mid X]</math> is ...or any martingale sequence whose diferences are bounded by a constant, the probability that it deviates <math>\omega(\sqrt{n})</math> far away from the starting p ...

* 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); ...

...f overall bias if the original numbers are positive or negative with equal probability. However, this rule will still introduce a positive bias for positive numbe ...f overall bias if the original numbers are positive or negative with equal probability. However, this rule will still introduce a negative bias for positive numbe ...