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
4k views

Why is ZPP = RP ∩ co-RP?

I am trying to prove the theorem that ZPP = RP $\; \cap \; co-RP$. If $L \in \; \subseteq RP \; \cap \; co-RP$ then I can see that it belongs to $ZPP$. But I am unable to prove the reverse direction, ...
6
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1answer
1k views

What does the “principle of deferred decisions” formally mean

I have encountered the phrase "Principle of deferred decisions" in Mitzenmacher and Upfal's book on Randomized Algorithms and several other courses online. Isn't it just conditional probability? In my ...
3
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0answers
82 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 ...
4
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1answer
704 views

Huffman Coding and Depth Calculation?

I'd like to as a variation on this question regarding Huffman tree building. Is there any theory or rule of thumb to calculate the depth of a Huffman tree from the input (or frequency), without ...
1
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1answer
116 views

Different probabilistic statement for Simple Uniform Hashing

Let me denote the number of elements with $n$ and the size of the table with $m$. I was trying to understand the Simple Uniform Hashing assumption that people and books describe in works and make them ...
2
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3answers
493 views

What is the formal analysis with Simple Uniform Hashing that the load factor is $\alpha = \frac{n}{m}$

Recall the definition of the load factor is the average number of elements in a chain for hashing for a table $T$ of size $m$ (with $n$ elements in consideration). Let $n_j = T[j]$ be the size of the ...
2
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1answer
133 views

Probability bounds on size of smaller partition in randomized quicksort

Let $0 < a < 0.5$ be some constant. We have an $n$-element array as input. Randomized quicksort chooses one element from array uniformly at random as a pivot and partitions. With probability $1-...
2
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1answer
567 views

Complexity of dynamic card game algorithm

Consider the following dynamic card game with a regular deck of 26 red cards and 26 black cards. A dealer draws the unturned cards one by one, and we can ask him to stop at any time. For every red ...
3
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1answer
146 views

A facility location problem

Consider the following scenario. There are N localities in a town where population for locality $L_i$ is denoted by $P_i$, $i \in {1,\ldots,n}$. We need to place K hospitals around the town in a way ...
-1
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1answer
75 views

Random quadtrees

I have $N$ uniformly distributed 2D points and I want to find out how many points lie in some small rectangular region. However, the number of points can be arbitrarily large (e.g., $N=10^8$), so I ...
1
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2answers
77 views

Is shuffling a set of items after popping an item meaningfully more random than doing it once, before starting?

I'm working on a thing to randomly assign people into a shift. There's mostly 2 sets of people, "free" and "assigned". Is shuffling the "free" set after assigning an employee meaningfully more random ...
9
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1answer
305 views

Pseudo-random sequence prediction

Disclaimer: I am a biologist, so sorry for (perhaps) basic question phrased in such crude terms. I am not sure if I should ask this question here or on DS/SC, but CS is the largest of three, so ...
3
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1answer
257 views

How do POMDPs and Dynamic Influence Diagrams differ?

To give some perspective, first consider the following diagram comparing Markov Chains, HMMs, MDPs, and POMDPs (I'm not sure who to credit for it). Fully observable ...
4
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1answer
57 views

One-shot Private Randomness Extractor

Suppose a pair of random variables $(X,Y)\in\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$ is given. I am interested in a deterministic mapping $f:\mathcal{Y}\to \{0, 1\}^k,$ for ...
2
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1answer
278 views

Why arrival process of packets at a switch is not a Poisson Process?

"Packet arrivals are not Poisson .... but some events are, such as web requests and new flow arrivals" I know since the network traffic is very bursty, they are not Poisson. But I am unable to ...
1
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1answer
75 views

Dempster-Shafer theory initial belief values

I am looking to implement D-S Theory in my (computer science) research, I'll be using it to determine the probability that a triggered sensor event is a true positive. How would you calculate an ...
4
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1answer
174 views

Balanced allocation-Hash table- overflow probability

My question is related to this: Hash-Table in Practice In [1] page 7, it is said that if we throw $n$ balls into $k$ bins, then each bin contains at most $\frac{n}{k}+O(\sqrt[2]{(\frac{n}{k})\log k}...
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 ...
3
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1answer
658 views

What is the best you can do with a noisy message?

You play a TV game in which you have to open one door out of $n$. Behind each door, there is a treasure with probability 1/2, independent of the other doors, so your apriori chance of winning is 1/2. ...
1
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1answer
80 views

Simulating continuous time semi-Markov state machine and changing transition probability on the fly

The problem that I'm trying to solve (well, I think that I almost did, but need a review from someone more experienced) is about changing probability of the transition for semi-Markov state machine ...
1
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1answer
46 views

Proving simple bound on coupon collector

I came across this paper which gives bounds on coupon colloector problem. Page 451 contains a table where reference to U1 is given as 'folklore'. I presume this is trivial to follow from the ...
4
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0answers
50 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 ...
3
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1answer
70 views

Maximum entropy probability distribution among Solomonoff priors

If we take Solomonoff's prior $m$, defined here and normalize it we get a probability mass function on all finite words. But, the pmf isn't completely determined until we fix a universal Turing ...
2
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1answer
185 views

Collisions in independent hashing

Let $H$ be a $s$-wise independent family of hash functions from $\{1,\ldots,M\}$ to $\{1,\ldots,N\}$. It is easy to bound one collision, but are there good bounds for muliple collision ?
1
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1answer
527 views

How would you implement truly random hash functions in practice?

Suppose that $[U] = [0,...,U-1]$ is the universe from which all elements will be taken, and $A$ a hash table of size $m$. A hash function $h:[U]\rightarrow[m]$ is truly random if For any set of ...
4
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2answers
263 views

Minimal number of attempts at a multiple choice exam needed in order to pass, without any prior knowledge

A test is consisted of $N$ multiple choice questions, each has $k$ possible answers. A test solution is the sequence of answers $S\in[k]^N$. Given is a black box which receives a solution as input ...
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1answer
74 views

Probabilistic Analysis in real time network : Error in data and feedback channel. Two users communication

I'm studying probabilistic analysis in real time network. We have learned how many attempts in average are required to transmit a packet when there is no error on the feedback channel. With no error ...
4
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1answer
139 views

Average vs Worst-Case Hitting Time

Consider a simple random walk on an undirected graph and let $H_{ij}$ be the hitting time from $i$ to $j$. How much bigger can $$ H_{\rm max} = \max_{i,j} H_{ij}, $$ be compared to $$ H_{\rm ave} = \...
2
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1answer
809 views

Probabilistic algorithm with two-sided error

I am currently studying probabilistic algorithms and came across three major complexity classes: BPP: worst-case polynomial time, two-sided error RP: worst-case polynomial time, one-sided error ZPP: ...
0
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1answer
53 views

Estimate distribution of a composite variable

Suppose I have N sets of numbers (10 numbers per set) {a1, ....., a10}. I form a sum by taking one number at random from each set. SUM = num from set 1 +......+ num from set N. If I do this a large ...
2
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1answer
548 views

How does derandomization of 3SAT work via conditional expectations?

Given a single SAT clause with its 3 literals coming from 3 different variables it is obvious that a random assignment of values will satisfy it with probability 7/8 But I do not understand how from ...
6
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1answer
851 views

Generate a random graph with geometrical degree distribution

I'm working on graph generation, trying to implement the RT-nested-Smallworld network model described in this paper. We are talking about generating an undirected graph in a slightly different way ...
2
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1answer
60 views

How does one change the probability bounds in probabilistic complexity classes without changing the class?

I see this theorem whose proof is not clear to me : "Let $L \subseteq \{0,1\}^*$ be a language and suppose that there exists a polynomial time PTM M such that for every $x \in \{0,1\}^*$ and $Pr[ M(...
2
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1answer
47 views

Can one use the PCP theorem to prove correctness of deternimistic algorithms?

I am thinking of the equality "PCP(O(log(n)),0) = P" Say I have a deterministic polynomial time algorithm $A$ whose correctness I can't prove immediately. But say I create a probabilistic version of ...
6
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1answer
762 views

Chernoff bound when we only have upper bound of expectation

If $X$ is a sum of i.i.d. random variables taking values in $\{0,1\}$ and $E[X]=\mu$, the Chernoff bound tells us that $$\Pr(X\geq (1+\delta)\mu)\leq e^{-\frac{\delta^2\mu}{3}}$$ for all $0<\...
3
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1answer
153 views

Analysis of sorting Algorithm with probably wrong comparator? [duplicate]

It is an interesting question from an Interview, I failed it. An array has $n$ different elements $[A_1, A_2, \ldots, A_n]$ (random order). We have a comparator $C$, but it has a probability p to ...
3
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1answer
210 views

“Practical forms” of Chernoff bound for inequality in expectation

From Wikipedia: The above formula is often unwieldy in practice, so the following looser but more convenient bounds are often used: (i) $Pr(X\geq (1+\delta)\mu)\leq e^{-\frac{\delta^2\mu}{3}}, ...
13
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2answers
702 views

Efficient algorithm to generate two diffuse, deranged permutations of a multiset at random

Background $\newcommand\ms[1]{\mathsf #1}\def\msD{\ms D}\def\msS{\ms S}\def\mfS{\mathfrak S}\newcommand\mfm[1]{#1}\def\po{\color{#f63}{\mfm{1}}}\def\pc{\color{#6c0}{\mfm{c}}}\def\pt{\color{#08d}{\mfm{...
5
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1answer
2k views

What do we know about $NP \cap co-NP$?

What do we need about the intersection of $NP$ and $co-NP$ apart from the fact that $P$ is a subset of it? (beyond what these answers here say, What do we know about NP ∩ co-NP and its relation to ...
2
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1answer
40 views

Return values of Probabilistic Programs

Syntax and semantics section of this paper on probabilistic programming mentions that the return expression of the program is a function f satisfying $ f : \sum \rightarrow \mathbb{R}^{\inf}_{\...
5
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1answer
66 views

Distribution of Ones in a Psuedorandom Sequence

Let S be a string in the set (0,1) produced by taking the AND of the output of two maximal length linear feedback shift registers of large period (say 128 bits). It's easy to see from the truth table ...
9
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2answers
3k views

What are Markov chains?

I'm currently reading some papers about Markov chain lumping and I'm failing to see the difference between a Markov chain and a plain directed weighted graph. For example in the article Optimal state-...
1
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1answer
302 views

Markov Chain w/ non-stochastic matrix

I've come across a problem which at first appeared to be a markov process however the transition matrix of the graph is non-stochastic. That is, the probabilities among edges leaving a node do not sum ...
5
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1answer
298 views

Random Graph is a good expander

If a (n,d) random graph is a n-vertex graph defined as : Choose d random permutations $\pi_1 \ldots \pi_d $, from [n] to [n]. Take edge (u,v) if $v = \pi_i(u)$ for some i. I am trying to prove that, ...
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1answer
770 views

Naive Bayes MapReduce

I have the following question to answer in a MapReduce assignment sheet The Naive Bayes classifier is a widely-used tool for analyzing data. Consider a data set that has $n$ data items, each of ...
0
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2answers
7k views

How to calculate probability of packet loss and drop rate?

In a queuing system (M/M/1) with a finite packet capacity $z$, how do you determine the probability of packet loss if we assume that packets are dropped when the system is full? Packets arrive with a ...
0
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1answer
201 views

Why is the most probable assignment for all variables in MRFs called MAP assignment?

I am new to graphical model, especially Markov Random Fields. I have a question about MAP assignment. Let say we have the graph structure and all the potential functions. MAP assignment in MRFs is ...
12
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0answers
377 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 $n$ ...
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1answer
615 views

Understanding polynomial equality testing using randomized algorithms

A file is downloaded from a server and is represented as $a = \{0, 1\}^n$. The server has that file as $b = \{0, 1\}^n$. We want to ensure a degree of certainty that $a=b$, using a randomized ...
2
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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 $p$ ...

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