Questions tagged [probability-theory]

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

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2
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1answer
21 views

How do you marginalize in graphical model elimination?

I'm reading Michael I. Jordan's book on probabilistic graphical models, and I don't understand the elimination algorithm presented in chapter 3. To narrow the question down, consider page 6. In ...
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1answer
96 views

Network throughput with random delay selected from uniform distribution

Background: I am working with IoT devices which broadcast status messages over a wireless channel periodically and at a rather high rate (500-5000 Hz). Receiving every message is not crucial but the ...
3
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1answer
166 views

Sampling numbers from a weighted set that sum to constant value

So I have a multi-set of positive integers $S = \{n_1, n_2, \dots\}$ with associated weights $W = \{w_1, w_2, \dots\}$. I want to sample some numbers, without replacement, from $S$ according to ...
2
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1answer
103 views

Bertrand's ballot theorem

I want to understand the dynamic programming equation of https://en.wikipedia.org/wiki/Bertrand%27s_ballot_theorem theorem. it is this If i number of people voted for A and j number of people voted ...
2
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1answer
50 views

Complexity/Hardness of a generalization of an Inclusion/Exclusion problem

I would appreciate some help in determining the complexity/hardness of an inclusion/exclusion problem described in Wikipedia: https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle#...
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1answer
33 views

Expectation of $u'^t v$ = $u^t v$

I have another question with dimensionality reduction. I have a matrix $S \in R^{k \times d}$ and S is in {$- \frac{1}{\sqrt k}, \frac{1}{\sqrt k}$} and i have two vector $u,v \in R^d $. I need to ...
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0answers
48 views

How to generate a random number in a given range with uniform probability [closed]

I have used programming languages which generate a random number in a given range. Let's say we have a range of 1 to 10 each number has the probability of 1/10 to get selected. What is the criteria ...
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1answer
312 views

Proof of an Infinite Binary Sequence

I have a problem where given an infinite binary sequence S ∈ {0, 1}∞ to be "prefix-repetitive" if there are infinitely many strings w ∈ {0, 1}* such that ww is a prefix of S. I need to prove that if ...
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0answers
33 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 ...
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0answers
68 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 ...
1
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1answer
41 views

Expectation for the number of leaf nodes in a randomized tree construction

Consider this procedure for building a tree from $v_1, v_2, ..., v_n$: insert $v_1$ insert $v_2$ and connect it to $v_1$ via a directional edge from $v_2$ to $v_1$ insert $v_3$ and with a uniform ...
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1answer
74 views

HyperLogLog leading zeros distribution

I'm reading this article. In particular I'm having some trouble trying to replicate the results of the first image in that article. For x=9, the graph says that the probability is 0.20 aprox. But ...
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1answer
27 views

Expectation of $\langle s,x \rangle^2$

I'm studying dimensionality reduction (SVD in particular), and I saw the following question: Assume we have a vector $x \in \mathbb R^d$, and consider $F(x)=s^t x$ , where $s$ is a $d$-...
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2answers
294 views

probability of collision

my data's range is from 1 to 9 and I have two subsets of integers from this range. the hash function takes each of this subsets and calculate product of these three integers and maps this set to the ...
1
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1answer
9 views

Is there a term for the distribution of numbers-of-digits of a bounded uniform distribution

Let $X$ be an integer sampled uniformly from the (integer) range $\{ 0...2^k - 1 \}$ . Now, consider the distribution of $\#\text{bits}(X) = \begin{cases} 1 & X = 0 \\\\ 1 + \lfloor \log_2(X) \...
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1answer
25 views

Confusion About Min-Cut Probabilities

Currently going through a video on Counting Minimum Cuts by Tim Roughgarden. $(A_{i},B_{i}) = \big((A_{1},B_{1}), ..., (A_{t},B_{t})\big) \forall i \in \Bbb{R}$ $P\big((A_{i},B_{i})\big) \geq \frac{1}{...
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1answer
131 views

Post-selection and complexity theory [closed]

I read about post-selection and didn't understand the meaning behind this thing. I didn't understand the Wikipedia article well, so what is a simple but understandable explanation of post-selection ...
1
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1answer
67 views

How many prefix code we can find for a given distribution of probability?

A source emits 5 signals s1, s2, s3, s4 and s5 whose probabilities are as follows: 1/3, 1/3, 1/9, 1/9, 1/9, 1/9. How many prefix codes we can construct on A={a, b, c}? And how many are there with the ...
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2answers
169 views

challenge: Closing hash, linear probing, probe length 1

The challenge says that there is a hash algorithm with a bug, the bug is given if it reaches the end of the array and did not find space to save the element, it discards it directly (that is, it does ...
1
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1answer
367 views

Is the bitwise-xor of a Uniform bit string and a non_uniform bit string Uniform?

Having two bit strings $x,y \in \left\{0,1\right\}^n$, where $x$ is selected following a uniform distribution but $y$ is not. Is $z = x \oplus y$ uniform?
3
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1answer
28 views

Variance of MAXSAT clause satisfiability

For a given MAXSAT problem, it is trivially easy to compute the mean number of clauses satisfied for all assignments, or equivalently the expected number of clauses satisfied by a random assignment. ...
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1answer
26 views

What is the relation between the quantity of information of $A_i$ and $A$, where $A=\bigcup_iA_i$?

Let $A$ be an event divided into 4 events $A_i$ with the same probability. Why does the quantity of information of $A_i$ satisfy $$ I(A_i) = I(A) + \log (4)?$$
3
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2answers
224 views

Uniform sampling over non-standard simplex

Uniform sampling over a $n$-dimensional standard simplex is described here: Uniform sampling from a simplex I want to sample one point from a non-standard simplex with vertices at: $s_{i}\vec{e_{i}}$...
1
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1answer
197 views

Given a rng that outputs 0 or 1 with an equal probability, make a rng that generates 1 with a probability p

So this was an interview question I had a few weeks back that I just haven't been able to think of how to solve... Given a random number generator that returns 0 or 1 with a 50% chance of either, ...
1
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1answer
47 views

Maximizing entropy under constraint

How do I prove that entropy is maximal for $P(A_2) = \cdots = P(A_n) = (1-a) /(n-1)$ while $P(A_1) = a$ (a fixed number) and $A_1,…, A_n$ is a partition of the sample space?
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1answer
50 views

Probability of randomly designated subsets cover the universe

Let $U=\{1,2,\ldots,n\}$ and $S \subseteq \mathscr{P}(U)$. Let $T$ be a subset of $S$, randomly constructed selecting independently each element of $S$ with probability $p$. Is there a polynomial ...
8
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1answer
275 views

Is it possible to simulate a fair coin with a finite number of tossing of a biased one?

It is a classic problem to simulate a fair coin with a biased one. According to Fair Coin (wiki), John von Neumann gave the following procedure: Toss the coin twice. If the results match, start over,...
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2answers
695 views

How to determine seed collision probability in a PRNG?

I want to use a PRNG to generate random patterns. I would provide the PRNG with a hash value as a seed. Ideally, the seed size would be 64-bit or 128-bit and I would expect no collisions if the seeds ...
2
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2answers
96 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
112 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=...
2
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1answer
191 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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1answer
48 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 $...
6
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3answers
160 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 ...
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0answers
117 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
69 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. ...
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0answers
82 views

Wald's equation for dependent decrements

I'm taking a course on Coursera about approximation algorithms for fun and one of the modules uses probability to analyze the cost of a linear program with randomized rounding solution to the set ...
1
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1answer
134 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
61 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 ...
3
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2answers
857 views

Why analysis of Aloha protocol uses Poisson distribution?

Pretty much in all of the analysis of the Aloha protocol that I read, it is assumed that the distribution of packet arrivals is Poisson. What is the rationale behind it? Isn't it actually binomial ...
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1answer
30 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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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 ...
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2answers
2k views

Probabilty that quicksort partition creates an imbalanced partition

I have come across this question: Let 0<α<.5 be some constant (independent of the input array length n). Recall the Partition subroutine employed by the QuickSort algorithm, as explained in ...
1
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1answer
136 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 ...
2
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1answer
24 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 ...
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0answers
385 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 $H$. Given a hash ...
1
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1answer
21 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
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1answer
615 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], \...
2
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2answers
65 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 ...
0
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1answer
39 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
116 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, ...

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