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Questions tagged [probability-theory]

Questions about the branch of mathematics concerned with modelling and analysing random phenomena.

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2
votes
1answer
58 views

The notion of density of distribution

I have difficulties in understanding the notion of density for distribution. Notion of density for distribution. A distribution $H$ over $\{0,1\}^n$ has density $\sigma$ if for every $x \in \{0,1\}^{...
6
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1answer
3k views

Negligible Function in Cryptography

In the field of Cryptography and Computation Complexity there is a notion of negligible function. I have some difficulties in understanding intuition behind this notion. The following are some ...
7
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1answer
686 views

Shannon Entropy to Min-Entropy

In many papers I've read that it is well known that the Shannon entropy of a random variable can be converted to min-entropy (up to small statistical distance) by taking independent copies of the ...
18
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1answer
8k views

Applying Expectation Maximization to coin toss examples

I've been self-studying the Expectation Maximization lately, and grabbed myself some simple examples in the process: From here: There are three coins $c_0$, $c_1$ and $c_2$ with $p_0$, $p_1$ and $p_2$...
5
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1answer
344 views

Mutual information and moment generating functions

I went to listen to a workshop and someone from the audience asked the presenter how the moments can improve the mutual information. I am learning about MI (Mutual Information) so didn't have enough ...
5
votes
1answer
2k views

Pointwise mutual information vs. Mutual information?

I am learning about information theory and mutual information. However, I am quite confused with MI(Mutual information) vs. PMI(Pointwise mutual information) especially signs of MI and PMI values. ...
5
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3answers
200 views

Unbiasing of sequences

There is the well-known method of unbiasing of bit sequences due to von Neumann. Are there similar schemes applicable to other sequences, e.g. the result of throwing a normal die?
0
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1answer
96 views

Smoothing frequencies without count data

I have frequency data for different events under two conditions, resulting in sets of frequencies F1 and F2. I would like to normalize the frequencies of events under condition 1 by their frequencies ...
1
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3answers
434 views

Random graph model

When we say that in random graph we add an edge with a certain fixed probability, what do we actually mean? For example if probability is 0.5, does that mean that we can just add two edges in a graph ...
12
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3answers
2k views

Discrepancy between heads and tails

Consider a sequence of $n$ flips of an unbiased coin. Let $H_i$ denote the absolute value of the excess of the number of heads over tails seen in the first $i$ flips. Define $H=\text{max}_i H_i$. Show ...
6
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1answer
162 views

Mental poker: proving dealt hand is fair

I have just read mental poker, described in this fascinating paper(PDF) by cryptographic greats Adi Shamir, Ron Rivest, and Leonard Adleman. Assuming I have a website, (TTP) how can I prove to the ...
3
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2answers
183 views

Condition in Shamir Secret Sharing Scheme

For Shamir's secret sharing scheme (doi 10.1145/359168.359176), one obtains a random polynomial $q$ of degree at most $n-1$ (over $\mathbb{Z}_p[x]$). The constant coefficient of this polynomial is ...
7
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2answers
213 views

Building probability distribution functions from observation

There are N players and M objects, each of the objects has a value. Each player has a strategy in choosing an object. Each round a player will choose an object, many players can choose the same object....
3
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1answer
443 views

How many random walks to start from each node?

Assume that we are given a real life graph, DBLP network in my case, where degree distribution of nodes follows a power law (many nodes have 1, 2 neighbors, and only a few nodes have hundreds of ...
2
votes
1answer
129 views

probability wheel, redistribution of probabilities

I have a contiguous ordered data structure (0 based index): x= [1/3, 1/3, 1/3] Let's say I selected index 1 and increased the probability by 1/3. Rest of the ...
33
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2answers
1k views

How asymptotically bad is naive shuffling?

It's well-known that this 'naive' algorithm for shuffling an array by swapping each item with another randomly-chosen one doesn't work correctly: ...
3
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2answers
236 views

Construction of binary random variable

We throw two coins in a row and thus get the event space $\{ZZ, WW, ZW, WZ\}$. Each of the 4 elementary events has a probability $1/4$. how can I construct 3 binary random variable $x_1$, $x_2$, $x_3$...
6
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3answers
126 views

Maximal derangements

When one shuffles playing cards, the goal is evidently to achieve a possibly big derangement of a given deck. For manual shuffling there are terms like inshuffle, outshuffle etc. I like to know ...
1
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1answer
1k views

Hiring one person out of n — rank and probablity

I am studying algorithms from CLRS book. I try to understand the difference between probability of hiring the $i$th person out of $n$ and probability of hiring the $i$th person out of $n$ persons ...
0
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3answers
58 views

Heuristically determine a value f such that a probability d/f approaches 1/2

We have a set X of N elements. We want to get a new set X' having a size M < N. ...
13
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1answer
15k views

Smoothing in Naive Bayes model

A Naive Bayes predictor makes its predictions using this formula: $$P(Y=y|X=x) = \alpha P(Y=y)\prod_i P(X_i=x_i|Y=y)$$ where $\alpha$ is a normalizing factor. This requires estimating the parameters ...
5
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1answer
2k views

Determining Probability from a Graph

Lets say I have node A that connects to 10 other nodes. 6 of those nodes have Property 1 and the other 4 have Property 2. How can I easily determining the probability of landing on a node with ...
0
votes
1answer
664 views

Time series probability and mutual information

There is a time series of say $100$ data points. I wish to assign symbols of $0, 1, 2$ for each unique data point. The issue is I have tried but got stuck since no matter I specify the symbols, the ...
3
votes
1answer
565 views

How to random-generate a graph with Pareto-Lognormal degree nodes?

I have read that the degree of nodes in a "knowledge" graph of people roughly follows a power law distribution, and more exactly can be approximated with a Pareto-Lognormal distribution. Where can I ...
21
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3answers
3k views

Is rejection sampling the only way to get a truly uniform distribution of random numbers?

Suppose that we have a random generator that outputs numbers in the range $[0..R-1]$ with uniform distribution and we need to generate random numbers in the range $[0..N-1]$ with uniform distribution. ...
8
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2answers
1k views

Does there exist any work on creating a Real Number/Probability Theory Framework in COQ?

COQ is an interactive theorem prover that uses the calculus of inductive constructions, i.e. it relies heavily on inductive types. Using those, discrete structures like natural numbers, rational ...
4
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1answer
101 views

Probabilities of duplicate mail detection by comparing notes among servers

I have the following problem: We want to implement a filtering strategy in e-mail servers to reduce the number of spam messages. Each server will have a buffer, and before sending an e-mail, it ...
11
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1answer
2k views

Number of clique in random graphs

There is a family of random graphs $G(n, p)$ with $n$ nodes (due to Gilbert). Each possible edge is independently inserted into $G(n, p)$ with probability $p$. Let $X_k$ be the number of cliques of ...
4
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2answers
623 views

Online generation of uniform samples

A source provides a stream of items $x_1, x_2,\dots$ . At each step $n$ we want to save a random sample $S_n \subseteq \{ (x_i, i)|1 \le i \le n\}$ of size $k$, i.e. $S_n$ should be a uniformly chosen ...
3
votes
1answer
439 views

Deterministic and randomized communication complexity of set equality

Two processors $A, B$ with inputs $a \in \{0, 1\}^n$ (for $A$) and $b \in \{0, 1\}^n$ (for $B$) want to decide whether $a = b$. $A$ does not know $B$’s input and vice versa. A can send a message $m(...
3
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1answer
103 views

Making random sources uniformly distributed

How do I build a random source that outputs the bits 0 and 1 with $prob(0) = prob(1) = 0.5$. We have access to another random source $S$ that outputs $a$ or $b$ with independent probabilities $prob(a)$...
5
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2answers
156 views

Extracting non-duplicate cells in a particular matrix with repeated entries

Consider a board of $n$ x $n$ cells, where $n = 2k, k≥2$. Each of the numbers from $S = \left\{1,...,\frac{n^2}{2}\right\}$ is written to two cells so that each cell contains exactly one number. How ...
7
votes
1answer
170 views

Prove fingerprinting

Let $a \neq b$ be two integers from the interval $[1, 2^n].$ Let $p$ be a random prime with $ 1 \le p \le n^c.$ Prove that $$\text{Pr}_{p \in \mathsf{Primes}}\{a \equiv b \pmod{p}\} \le c \ln(n)/(n^{...
20
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1answer
497 views

Algorithm to chase a moving target

Suppose that we have a black-box $f$ which we can query and reset. When we reset $f$, the state $f_S$ of $f$ is set to an element chosen uniformly at random from the set $$\{0, 1, ..., n - 1\}$$ where ...
14
votes
1answer
7k views

Randomized Selection

The randomized selection algorithm is the following: Input: An array $A$ of $n$ (distinct, for simplicity) numbers and a number $k\in [n]$ Output: The the "rank $k$ element" of $A$ (i.e., the one in ...
16
votes
2answers
591 views

How does variance in task completion time affect makespan?

Let's say that we have a large collection of tasks $\tau_1, \tau_2, ..., \tau_n$ and a collection of identical (in terms of performance) processors $\rho_1, \rho_2, ..., \rho_m$ which operate ...
3
votes
1answer
163 views

Is it viable to use an HMM to evaluate how well a catalogue is used?

I was interested on evaluating a catalogue that students would be using to observe how is it being used probabilistically. The catalogue works by choosing cells in a temporal sequence, so for ...
23
votes
5answers
8k views

How to approach Vertical Sticks challenge

This problem is taken from interviewstreet.com We are given an array of integers $Y=\{y_1,...,y_n\}$ that represents $n$ line segments such that endpoints of segment $i$ are $(i, 0)$ and $(i, y_i)$. ...
6
votes
4answers
3k views

What are the uses of Markov Chains in CS? [closed]

We all know that Markov Chains can be used for generating real-looking text (or real-sounding music). I've also heard that Markov Chains has some applications in the image processing, is that true? ...
26
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9answers
17k views

Generating uniformly distributed random numbers using a coin

You have one coin. You may flip it as many times as you want. You want to generate a random number $r$ such that $a \leq r < b$ where $r,a,b\in \mathbb{Z}^+$. Distribution of the numbers should ...