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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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Question on preventing k from reducing too quickly during KMV intersection

This question considers KMV, an algorithm that is able to estimate the cardinality (unique item) from a stream of data. The way it does it is to first map the stream of data to a space that almost ...
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1answer
42 views

How do we prove the time complexity of this simple problem in probabilistic inference on a Bayesian network?

Perhaps a rather trivial question, but I'm trying to refresh on proof methods in CS... Suppose we have a simple Bayesian network with two rows of nodes: $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots,...
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42 views

Derive the expected # of steps that are taken to perform all the operations (n operations) [duplicate]

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($T, i$) ...
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1answer
76 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($T, i$) ...
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2answers
86 views

expected length of linked list

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($T, i$) ...
2
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1answer
20 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
34 views

Hitting probability of random walk within given number of steps

Given m,n dimensions of a 2D matrix; (i,j) initial co-ordinates; (x,y) final co-ordinates. What is the probability of being at (x,y) after at most k steps if we start from (i,j) initially? We can ...
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1answer
57 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 ...
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17 views

Probability in 1-universal hash function

I am trying to prepare for an exam and I am not sure how to solve this task: Given is a hash function with m buckets, which uses a 1-universal hash function h: U -> H and handles collisions with ...
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1answer
36 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 ...
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1answer
37 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 ...
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1answer
28 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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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
59 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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27 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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43 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 ...
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1answer
28 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
29 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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1answer
30 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
21 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
76 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 ...
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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
29 views

How can I minimize floating point error when multiplying normal distribution PDFs?

If you multiply two normal distribution PDFs with means $\mu_1$ and $\mu_2$ and variances $v_1$ and $v_2$, then according to this page, the new mean is $$\mu = \frac{\mu_1 v_2 + \mu_2 v_1}{v_1 + v_2}$...
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1answer
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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
138 views

expected pairwise square euclidean distance between points

How can I show that the expected pairwise square euclidean distance between points in $X$ is $Θ(d)$? Where $X$ is a $(x_1,...x_n)$ of points generated uniformly at random in the unit, d is d-...
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1answer
13 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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1answer
54 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?
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1answer
14 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
20 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 $$ I(A_i) = I(A) + \log (4)?$$
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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: $s_{i}\vec{e_{i}}$...
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1answer
113 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, ...
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1answer
39 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
42 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 ...
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2answers
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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 ...
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1answer
126 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 ...
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2answers
144 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 ...
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2answers
41 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 ...
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2answers
63 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
45 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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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 ...
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1answer
32 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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1answer
48 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 ...
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1answer
51 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 ...
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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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19 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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47 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.) ...
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1answer
44 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
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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 ...
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1answer
23 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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141 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 $H$. ...