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

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35 views

Probability that a random graph will remain planar after adding an edge

According to this answer, a random graph on $n$ vertices is a graph which has each of the $n\choose2$ edges independently with probability $1/2$ each. The probability of at most $3n-6$ edges (which is ...
4
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2answers
46 views

Fast sampling from discrete space

Assume we are given a set $X = \{x_1,...,x_n \}$ of size $n$, and a probability distribution $P$ over $X$. I am interested in an algorithm $A$ which can sample from $X$ according to $P$, i.e. $\Pr(A=...
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1answer
35 views

number of random sets needed to generate subset

Let $A\subseteq \{1\ldots n\}$ with $|A|=\alpha n, 0<\alpha\leq1$. Now we start generating random sets $B_i \subseteq \{1\ldots n\}$ with $|B_i|=\beta n$ where $0<\beta\leq\alpha$. How many $...
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0answers
29 views

distinguishing two biased coins

I had a simple probability question: suppose we have two coins, coin 1 is heads with probability $= 10\epsilon$ and coin 2 is heads with probability $=\epsilon/10$. Given an unknown coin, how many ...
2
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1answer
24 views

What is the average size of a jump in a binary tree?

Assume that a full binary tree is layed out in memory recursively in the following way. First the Root followed by Tree representation of left subtree followed by tree representation of right subtree. ...
1
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1answer
21 views

Question on word probability for hierarchical softmax used in natural language processing

I am reading the following paper: https://papers.nips.cc/paper/5021-distributed-representations-of-words-and-phrases-and-their-compositionality.pdf On page 4 of the paper they describe the ...
1
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1answer
42 views

Connection between probabilistic algorithm and distributional algorithm

A basic question about the connection between two notions. I am sure these are known notions in CS but I struggle with the basics: First Definition: A probabilistic algorithm $A(n)$ decides $L$ if for ...
1
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1answer
22 views

Probabilisitc timed automaton

I am kind of new to timed automaton domain. I am trying to understand in which way they are different to Markov Decision Process. First I know there objectives is to solve the non-determinism of a MDP....
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0answers
15 views

Conditional probability in Bayesian network

I have the following Bayesian network: I want to state that P(S|B) = 1 - P(!S|B) In the solution they use "P(S|B) = P(S|B,F) + P(S|B,!F)", which I understand but I don't understand why can't I use ...
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0answers
17 views

Optimality of Bayes Classifier

I know this question has been asked a few times but I find it hard to understand the solutions. Say, here (For example, I don't understand why the true error is defined in that answer the way it is.) ...
1
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1answer
33 views

Uniform sampling with constraints

Suppose one wants to uniformly sample a string $w$ of a given length over a finite alphabet, such $w$ satisfies a set of structural constraints (such as - "the third character has to be equal to the ...
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0answers
5 views

Relevance of Pure ALOHA for small network load G?

The theoretic performance of Pure ALOHA networks is a vastly studied topic and references can easily be found , e.g. on https://en.wikipedia.org/wiki/ALOHAnet#cite_note-tann-14 Using a Poisson ...
2
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1answer
22 views

Random award question

I'm assisting with the design of an algorithm over the next week to fit the following use case: A person walking in a store has a tablet and approaches possible customers to notify them of a ...
0
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0answers
23 views

Proving a hash function family is weakly universal

$H$ is a family of weakly universal hash functions if for two elements $x,y$: $$ P(h(x) = h(y)) \le \frac1m, $$ where $m$ is the size of the domain, and $h$ is chosen at random from $H$. ...
1
vote
1answer
18 views

Upper bound derivation of expected time for finding a large cut

Given an undirected graph $G$ with $n$ vertices and $m$ edges, we build a cut as following. Initially sets (of vertices) $A$ and $B$ are empty. For each vertex $v$ we flip a fair coin and according to ...
2
votes
1answer
47 views

Average-case analysis of linear search given that the desired element appears $k$ times

The problem below is adapted from CLRS Problem 5-2 "Searching an unsorted array": Consider a deterministic linear search algorithm which searches an array $A$ for $x$ in order, say $A[1], A[2], \...
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1answer
25 views

Markov Model to compute the probaility on the $n^{th}$ day

This is a question about Markov Models. Let's say we have the following situation Let's say that we want to find the probability that $2$ rainy days follow a nice day. You'd simply have $0.25 \cdot ...
2
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1answer
46 views

Avoiding correlated values and reduced collision resistance in multiple hash states

I need to design a hash function that fits this criteria: Limited to 32-bit integer operations (for compatibility with JavaScript bitwise operations, my target system) Produces an n-bit hash digest, ...
0
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1answer
15 views

Relation between size of hashtable and number of values to keep expected number of collisions below/equal to 1

This is an exam question from my algorithms and data structures course. You imagine an hash function with h: U->{0,..m} (this is from the original question, but i think m-1 would be correct) and n ...
2
votes
2answers
52 views

Why does the distribution of pseudorandom numbers change when operating like this?

We were introduced to the standard C rand() function, and how to use it. At some point during the class we were asked how to roll a die. The simple answer was to ...
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0answers
14 views

Presentation topic for computational solutions to problems involving probability

I'm looking for interesting applications of computing science for finding the probability of otherwise difficult problems. This is for an undergraduate level presentation in a probability course of ...
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1answer
30 views

Applying a Chernoff bound with Only an Upper Bound of the Expectation

First, I am aware at least one or two similar questions have already been asked on stack exchange, but I've gone through the answers they got and didn't find one that was satisfactory for my case. The ...
0
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1answer
83 views

ALOHA - Throughput and probabilities

I have a few questions regarding slotted-ALOHA. Assume a network have 25 users and transmission request probability = 0.25. 1) What is the throughput and what is the probability that a user will ...
5
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1answer
332 views

Reservoir sampling algorithm probability

I'm reading about the reservoir sampling technique called Algorithm R. The idea is we can take a sample of size $n$ from a population of size $N$ even when $N$ is unknown/too expensive to retrieve in ...
0
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1answer
46 views

Trying to understand CLRS bucket sort analysis

I'm trying to understand the analysis of bucket sort in CLRS. Specifically, equation 8.2 that states: $$ E[{n_i^2}] = 2 - \frac{1}{n} $$ To prove, CLRS: Random variable denoting number of elements ...
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1answer
66 views

Expected number of vertices with degree 2

A simple graph with $n$ vertices is constructed by randomly and independently placing an edge between every two vertices with probability $p$. What is the expected number of nodes with degree two? I ...
2
votes
1answer
47 views

Viterbi Algorithm: initial state with ONE probability

The Viterbi Algorithm can be used to calculate the most likely path, based on observations in a Hidden Markov Model. Using the same notations as Wikipedia, "each element ...
1
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1answer
126 views

Probability of k-clique in a random graph

I need to find the order of the minimum k = k(n) such that the probability of having at least 1 k-clique in a random graph $G(n, \frac{1}{2}$) is $\mathcal{O}(\frac{1}{n})$. $X_k$ is the random ...
2
votes
2answers
207 views

how to bound the probability that quicksort takes greater than n lg n time?

I am working on exercise 12.4-5 of CLRS (Cormen et al, Intro to Algorithms 3rd ed) Consider RANDOMIZED-QUICKSORT operating on a sequence of n distinct input numbers. Prove that for any constant k > ...
2
votes
1answer
110 views

How to estimate how many assignments satisfy a given DNF formula using Monte Carlo?

Admittedly, homework. For the purpose of this question: $\phi$ is a DNF formula similar to this one: $(x_1 \wedge \neg x_3 \wedge x_4) \vee (\neg x_1 \wedge x_2)$ Also $C_i$ is a clause in this ...
6
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1answer
2k views

Why are forks in the Blockchain eventually resolved?

I'm reading Wattenhofer's The Science of the Blockchain. On page 87, he states the following thoerem: Theorem 7.22. Forks are eventually resolved and all nodes eventually agree on which is the ...
3
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1answer
26 views

Polynomial Computation of the probability of a number of independent events

Suppose to have $n$ independent events $E_1, E_2,..., E_n$, where the probability of occurrence of event $E_i$ is $p_i$ (i.e., each event has its own probability of occurrence). We can easily define ...
2
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0answers
36 views

How do I do a biased shuffling of a deck of cards?

I currently have a deck of cards (in fact this deck is an array of sorted elements in descending order as indicated by the picture above), that I want to shuffle. However, the caveat is that I want ...
0
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0answers
33 views

Pathological input for a completely random hash function

I have a hash function $h$ (assume it's completely random, an element of a Universal family) which maps integers into one of $k=10$ groups. Now I'm looking for an input sequence of length $m$ for $h$ ...
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0answers
47 views

Average solutions in infinite search tree

I have the following problem: Consider a balanced infinite search tree with branching number $κ$. Consider a search problem with solutions which may be located at any node of the tree. The ...
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0answers
22 views

How are Probability and Non Determinism Related ? Alternatives to handle Non determinism?

I have been thinking about Non-determinism in any kind of state machine. Since I work on machine learning, I tend to think that probability is a means of handling non-determinism. Instead of ...
2
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0answers
28 views

Why can optimality be preserved when inserting a new conjunct into an optimally ordered conjunction of conditions? [duplicate]

In a programming language with short-circuiting, a conjunction of N independent conditions has the following expected cost: where: ...
2
votes
2answers
114 views

The optimal way to reverse engineer a binary classification problem

I am not sure if this question is more suitable for CS, theoretical CS or math so feel free to improve the description and migrate it. In a scenario very similar to popular binary classification ...
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0answers
147 views

How to choose value of additive smoothing in naive Bayes and why a higher value gives bad accuracy?

In Naive Bayes we often do additive smoothing as a fail safe. Consider the following expression: Lets say $$P(X_i) = \frac{count(X_i) + \alpha}{\sum_i^n count(X_i) + \alpha*total\_size}$$ How to tune ...
1
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0answers
30 views

Game with Random Digits (Markov Chain / Coupling)

I've been self-studying Markov Chains and came across a problem online here: http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf I'm not asking for anything too formal (I'...
2
votes
1answer
70 views

Confuse on using bucket number to replace hashing multiple times in hyperloglog explanation

In this blog the author states that the following So we now have a rather poor estimate of the number of values in the dataset based on bit patterns. How can we improve on it? One ...
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0answers
80 views

Probability in Dynamic frame aloha

If I've RFID Collision Arbitration System that is based on multi-frame dynamic frame ALOHA. How can compute the probability of the average tag resolution $L_n$? ,$P_s$ is probability for packet ...
3
votes
2answers
53 views

Are typical sets larger, when information is messier?

Let $0\le q<p\le \frac{1}{2}$, and let $P,Q$ be two Bernoulli Random Variables such that: $$Pr[P=1]=p ; Pr[P=0]=1-p$$ and $$Pr[Q=1]=q ; Pr[Q=0]=1-q$$ My question: Does it follow that, for any $\...
3
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1answer
68 views

Probability of k-connectedness for a random graph with given degree sequence

For a degree sequence $(d_1,\ldots, d_n)$ with $\min d_i \geq k$ pick a graph with that degree sequence uniformly at random. What's the probability that the graph is k-connected? I know that for ...
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1answer
35 views

Probabilistic analysis of finding 2nd largest value in vector

Given the following algorithm for finding the second largest element in a vector: ...
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0answers
23 views

$BPP$ class and theorem

Let $L \in BPP$ be decided by a poly-time $TM M (x, u)$ using certificates of poly-length $p(n)$. Then for every $n \in \mathbb{N}$ there exists a certificate $u_n$ s.t. for all $x$ with $|x| = n$: $...
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1answer
83 views

Equivalence of machine $M$ in $BPP$ and $EXP$

Ok, the proof is clear. But, if we prove that $BPP \subseteq EXP$ we should show that for every machin in $BPP$ there exists equivalent machine in $EXP$. I cannot see how the $EXP$-machine is ...
3
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1answer
39 views

Randomized BST height analysis : How $Z_{n,i}$ and $Y_{k-1}$ are independent?

I am referring to this video https://www.youtube.com/watch?v=vgELyZ9LXX4 at 1:08:39 . $n$ : number of nodes in the tree $Z_{n,k}$ : Indicator random variable that activates when rank of the root ...
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0answers
18 views

Probability that the word is of language using N-Grams

Some programm should allow users to identify the word language probability. All i have is some statistics - the frequency of using bigrams and ...
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0answers
41 views

Bound covariance of two discrete random variables

Let $X,Y$ be two random variables over a discrete probability space, such that $X \in [0,1]$ and $Y \in [0,1]$. I want to prove that $$ |\text{Cov}[X,Y]| \leq \sqrt{0.5 \; I[X,Y]}$$ where $I[]$ is ...