Questions tagged [probability-theory]

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

79 questions with no upvoted or accepted answers
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13
votes
0answers
385 views

Choosing a subset of binary variables to maximize the sum of the highest $K$

Consider the following problem: Input: integers $n > m > k$; $n$ numbers $0 \leq p_1, \ldots, p_n \leq 1$; $n$ numbers $r_1, \ldots, r_n$ where ($r_i \geq 0$). Let $X_1,\dots,X_n$ be $n$ ...
11
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0answers
1k views

Alternative to Bloom filter for extreme parameters

A Bloom filter is a space-efficient probabilistic data structure to perform membership-tests on a set (see Wikipedia's page for a definition; I use the same notations below). I am interested in a ...
8
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0answers
122 views

Compute the expected size of an approximation of vertex cover

Consider the following randomized approximation algorithm of vertex cover: Input: A graph G = (V, E). Output: A set $C_G \subseteq V$ a vertex cover of $G$. The algorithm: Set $C_G := \emptyset$. ...
6
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0answers
160 views

Correctness of a greedy Algorithm on Knockout Tournaments

You are given a function $\operatorname{rk}:\{1\dots 2^k\}\rightarrow \mathbb{N^+}$ representing the ranks of the players $1\dots2^k$ in a participating in a tournament. The tournament evolves in a ...
4
votes
1answer
58 views

Understanding simulated annealing information theoretically

So I recently rediscovered simulated annealing though a path that others seem to be well aware of. I was aware of Metropolis-Hastings as a sampling algorithm that creates a Markov-Chain who's ...
4
votes
0answers
61 views

Peculiar MCMC sampling problem

I have two random variables, X and Y, and Y is a positive real number. I can sample from $p(y|x)$, but I need to sample from $p(x)$, which I know to be proportional to $\frac 1 {E[y|x]}$. I could ...
4
votes
0answers
324 views

Mean and variance of number of buckets of length $i$ in hashing with chaining

Consider a hash table with $m$ buckets, with chaining as collision resolution policy. Given the set $S$ that will be stored in the hash table, let $X_i$ be the number of buckets whose chain length is $...
4
votes
0answers
52 views

Infer probabilities, for concatenation of words

Fix an alphabet $\Sigma$, and a set of words, $W = \{w_1,\dots,w_n\} \subseteq \Sigma^*$. I have a randomized model that works like this: Alice generates a random sequence of words, using some ...
4
votes
1answer
478 views

How to find a 2-wise independent hash family that is not 3-wise independent?

I'm trying to find a family of hash functions mapping $\{1, 2, ..., 2^n\}$ to $\{0, 1\}$ that is 2-wise independent but not 3-wise independent. Any ideas on that? I know two 2-wise independent ...
4
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0answers
60 views

Adversarial bin packing

An adversary gives you a set of items whose total size is $x$ (he gets to choose how $x$ is distributed. e.g. there may be $k-1$ items of size $\frac{x}{k}$ and 2 items of size $\frac{x}{2k}$). The ...
3
votes
0answers
44 views

Node depth in randomly built binary search tree

It can be proved that randomly built binary search trees of size $n$ are of depth $O(\log n)$ and it is clear that level $k$ has at most $2^{k}$ nodes (root's level is 0). I have an algorithm that ...
3
votes
0answers
54 views

Probability of colisson for classes of hash functions

I am going through some old exams in one of my courses, and I don't have access to solutions. I've found a problem which I am not sure how to tackle. I am not looking for the answer but some help/...
3
votes
0answers
59 views

Probability of a double cycle in cuckoo graph

I have read Chater 17. Balanced Allocations and Cuckoo Hashing in Mitzenmacher. Upfal. Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis and got ...
3
votes
0answers
83 views

Output process of a G/M/1 queue

What is the output distribution of a GI/M/1 (general input process and exponential service times) queue. The GI/M/1 is according to Kendall's notation: arrivals are independent but we do not know the ...
3
votes
0answers
63 views

Should Expectation Maximization take into account the Naive Bayes' independence assumption?

Should the independence assumption on which the Naive Bayes (NB) classifier is based, be taken into account when applying Expectation Maximization(EM) to infer missing values? The Naive Bayes ...
2
votes
0answers
34 views

Integer sampling with exponentially decreasing probability

Given a probability $p$ and an integer $N$, I would like to generate a sample $S$ of the population $P=\{0,1,...,N\}$ such that integer $m\in P$ is sampled with probability $p^m$. It is trivial to do ...
2
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0answers
30 views

In Universal Hashing- probability of poor performance is small and is the same for any set of keys of the same size

I was going through the text Introduction to Algorithms by Cormen et. al. where I came across the following claim: Universal Hashing uses randomization.For the example of a compiler's symbol table, ...
2
votes
1answer
22 views

Why there is no polynomially large sequence of polynomial large weights that derandomize the isolation lemma?

I was studying the paper Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size by Arvind and Mukhopadhyay and came across the following claim (Observation 1.2 on page 3): "More ...
2
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0answers
30 views

The No-Free-Lunch Theorem and K-NN consistency

In computational learning, The NFL theorem states that there is no universal learner. For every learning algorithm , there is a distribution that causes the learner to output a hypotesis with a large ...
2
votes
0answers
41 views

Correctness of Karger's min-cut Algorithm

tl;dr in the analysis for Karger's min-cut, the probability of an edge being in the min-cut in the $j$th iteration, $\frac{k}{0.5k(n-j)}$, neglects the fact that all the edges between the two ...
2
votes
0answers
113 views

What would be the probabilty of a randomly generated tree to be a Red-Black Tree

The question is not related to the homework I was working on a homework, and the specification was to generate a random tree with n elements(n being in the thousands for the assignment) and asked me ...
2
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0answers
62 views

Box labelling game

I have a box of stickers. It contains $n$ stickers. Each sticker is labelled with a different number from $\mathbb{Z}$. I have infinite supply of boxes. Box labelling game: I pick a random sticker ...
2
votes
0answers
136 views

Universal lower semicomputable semimeasure and Coding Theorem

I'm following Li and Vitanyi's book "An introduction to Kolmogorov complexity and its applications" 3ed. I'll rewrite here the definitions I need for my question. The authors define the reference ...
2
votes
0answers
97 views

Optimal wagering to minimize expected time to reach a target payoff

Suppose for simplicity we start off with starting amount $S = 1$ and we wish to reach target amount $T$. To do this we sequentially wager a certain amount and then win that amount with probability $p$ ...
2
votes
0answers
41 views

How to find number of occurences of specific distances in binary (search) trees?

I want to calculate the amount of tree structures that have a given maximal distance between two nodes given an amount n of nodes (or keys). E.g. with ...
2
votes
0answers
73 views

Which component sizes do we observe while randomly deconstructing a tree?

Suppose I have a connected graph with $n$ vertices and $n−1$ edges, that is in form of a tree. Now, I will add the number of vertices in the tree and uniformly randomly select a vertex. I break the ...
2
votes
0answers
154 views

Exact Inference in Bayesian Networks

I'm doing some exam study and came across a question I'm not really sure on. Consider the Bayesian network below: Let's denote "Disease" with $D$ and "Symptom" with $S$. I want to find $P(D \mid S_A,...
1
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0answers
18 views

Algorithm to mapping given probabilities to empirical probabilities

Consider following problem statement: You have given $n$ actions. You can perform any of them. Each action gives you success with some probability. The challenge is to perform given finite number of ...
1
vote
1answer
64 views

Count subset divisible by 3

I'm trying to solve this puzzle but I get stuck. I thought about trying to use the law of total probability to solve intermediate problems with subset of size $k$ but it didn't helped me that much. Is ...
1
vote
1answer
35 views

Show that if $\mathcal{H}$ is PAC learnable in the standard one-oracle model, then $\mathcal{H}$ is PAC learnable in the two-oracle model

This is a question $9.1$ from Understanding Machine Learning Chapter 3. It goes like this: Consider a variant of the PAC model in which there are two example oracles: one that generates positive ...
1
vote
0answers
19 views

Calculate the probability of undetected errors for the whole file (not packets) in the CRCs

If we have a large file, I want to know how this will affect the probability of undetected errors, especially in CRCs. I know that undetected error rate (packet or chunk) is BitR * BER * 0.5k where k ...
1
vote
0answers
39 views

How to estimate the number of elements inserted to a Bloom filter

A Bloom filter is a probabilistic data structure that allows encoding sets with false positives. Parameterized by the number of bits $m$ in the array $A$ (initialized to zeros), and number of hash ...
1
vote
0answers
59 views

Variance of chain length in hashtable

I have a hashtable with length $m$. Initially it's empty. Next, $n/2$ unique random numbers $\in [0, n]$ are added to it. What would be variance of chain length when such $n/2$ numbers are being added ...
1
vote
0answers
198 views

Channel coding and Error probability. Where are these probabilities from?

From where are the following probabilities? We consider BSCε with ε = 0,1 and block code C = {c1, c2} with the code words c1 = 010 and c2 = 101. On the received word y we use the decoder D = {D1,D2} ...
1
vote
0answers
48 views

Calculate probability in graphical model

I have the following graphical model, in which I wish to compute $p(Intelligence = 1|Letter = 1, SAT = 1)$ But I'm not sure how to rewrite $p(Intelligence = 1|Letter = 1, SAT = 1)$? I was told to ...
1
vote
0answers
45 views

Decomposition of Mutual Information

I came across a book where the author uses the following property of mutual information: Let $X$,$Y$,$Z$ be arbitrary discrete random variables and let $W$ be an indicator random variable. $$ (1)\ \ ...
1
vote
1answer
51 views

What kind of bigram probability smoothing is this?

I hope it isn't off topic but I need to understand this example. Given the corpus 12 1 13 12 15 234 2526 and smoothing factor of ...
1
vote
0answers
56 views

Probability of string misidentified in Bloom filter

I'm attempting a question related to Bloom filters: Our Bloom filter uses $3$ different independent hash functions $H_1, H_2, H_3$ that each take any string as input and each return an index into a ...
1
vote
1answer
160 views

Deriving the expected number of steps that is taken to perform the k'th operation

Consider a datatype whose objects will be sequences of elements that has the following two methods prepend($x, T$) which will insert an element to x to the beginning of the sequence T search($T, i$) ...
1
vote
0answers
32 views

Probability lower bound on a double cycle on two vertices in random cuckoo graph

I have read Chater 17. Balanced Allocations and Cuckoo Hashing in Mitzenmacher. Upfal. Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis and got ...
1
vote
0answers
88 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 ...
1
vote
0answers
429 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
vote
0answers
61 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'...
1
vote
0answers
132 views

Probability of finding the maximum element in a heap

You are given a minimum heap, with probability going to left is 50% and going to right is 50%. What is the probability that You will land up on a maximum element in the heap? For this scenario since ...
1
vote
0answers
24 views

Minimizing $E_{out}$ theoretically in linear regression

Related to linear regression with noise I've been given the following functional, depending on the function $h$: $$ E_{out}(h) = \int \int (h(x)-y)^2 dx dy$$ I want to prove that the function that ...
1
vote
0answers
129 views

Calculate expected values of “Craps Game” with the help of Prism Model Checker

I have modelled the craps game (https://en.wikipedia.org/wiki/Craps#Rules_of_play) as a dtmc with the prism model checker: ...
1
vote
0answers
67 views

Approximate conditional entropy

Given a set of random variables $X = \{x_1, x_2, \dots, x_n\}$. If the conditional entropy for all $Y \subset X - \{X_i\}$ where $|Y| \leq 5$. How to approximate conditional entropy when $|Y| = 10$ ...
1
vote
0answers
230 views

EM-algorithm for categorical hidden variables

I have the following model: Let's say two indepentent weighted six-sided dice $X$ and $Y$ with unknown probabilities (i.e. probability of 1 is unkown etc) and they have not necessarily the same ...
1
vote
0answers
101 views

Find minimum conditional entropy

Task : Given $X$ random variables. Find out the minimum conditional entropy for a variable $x_i \in X$ when $x_i$ is conditioned upon any combination $k$ remaining variables. Find $min(Entropy (x_i | ...
1
vote
0answers
59 views

probability that the vertex set {1,…,k} is component of random graph

Consider a graph with vertices 1,...,n and suppose that each of the $\binom{n}{2}$pairs of vertices is, independently, an edge of this graph with probability p.Let $P_n$ denote the probability that ...