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

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0
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6 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 | ...
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0answers
8 views

Does marginalizing on a Bayesian network preserve its original independence assumptions?

I know that marginalizing over a Bayesian network causes changes to the graph (e.g. marginalizing node c in the V-structure given by $a \rightarrow c \leftarrow b$ results in $a$ and $b$ being ...
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0answers
21 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 ...
1
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2answers
96 views

The transition function in a Markov decision process

A Markov decision process is typically described as a tuple $\langle A,U,T,R \rangle $ where $A$ is the state space $U$ is the action space $T: A \times U \times A \mapsto [0,\infty) $ is the state ...
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0answers
23 views

Expected value of exponential r.v [on hold]

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$ and let $c$ be a nonnegative constant. I want to answer the following questions: ...
4
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0answers
35 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 ...
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0answers
50 views

Problem with update in Dynamic Bayesian Networks

Consider the following Bayesian network: I want to impose constraints that state that a node can only be true (1) if at least one of its parents are true (1). So, for node $C$, the constraint takes ...
3
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1answer
109 views

Reconstructing a screen of permuted pixels

Reconstructing a screen of permuted pixels Summary Given a video with the pixel locations randomly permuted (once, for the entire video), can we (efficiently) reconstruct the original picture? Let: ...
6
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1answer
75 views

Two definitions of universal hash functions

I have seen two definitions of universal hash functions in the literature. For any $i \geqslant 2$ let $[i]=\{1,\ldots,i\}$. Definition 1: A family $\mathcal H$ of hash functions from $[n]$ ...
0
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0answers
24 views

Probability of producing a correct result at time t

A system with 3 nodes, as a function of time the probabilities of each of the 3 nodes producing a correct results are 0.99, $e^{(-t/10000)}$, $e^{(-t/1000)}$, respectively. How would you use the nodes ...
0
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2answers
33 views

Probabilities, Unigram and Bigram [closed]

Assume that we have these bigram and unigram data:( Note: not a real data) bigram: #a(start with a) =21 bc= 42 cf= 32 de= 64 e#= 23 unigram: # 43 a= 84 b=123 c=142 f=161 d=150 e=170 ...
1
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1answer
61 views

Markov Chain Mixing Time of the Complete Graph

I'm having a hard time understanding mixing time for Markov Chains on Complete Graphs (Kn). We can define the probability matrix for Kn where Pi,j=probability of going from i to j (technically ...
4
votes
1answer
52 views

Summation of a Random Variable

I'm having trouble grokking Cormen's Algorithms book on pg. 120. The context is the hiring problem (as it is called in the book). Here is the part that is tripping me up: "Let X be the random ...
1
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1answer
49 views

Expectation Maximization Algorithm for simple naive Bayesian network

I am trying to understand the following network A has two children - B & C (aka common cause) All the variables are binary and can be either 0 or 1. In data values are missing only for some ...
10
votes
1answer
93 views

What is the chance that this code terminates?

I wrote this Python code, and wondered if it sometimes simply doesn't terminate (assuming we had infinite memory/time and no recursion depth limit). Intuitively you'd think it terminates, since at ...
1
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1answer
42 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, ...
3
votes
1answer
61 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 ...
4
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0answers
42 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 ...
4
votes
1answer
116 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
36 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
66 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 ...
1
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1answer
59 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 ...
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1answer
103 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 ...
4
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1answer
71 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 ...
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1answer
40 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
62 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 ...
10
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1answer
199 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, ...
3
votes
1answer
53 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
votes
1answer
40 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
votes
1answer
49 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
vote
1answer
23 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
votes
1answer
51 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 ...
6
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0answers
282 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
655 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
38 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
41 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 ...
3
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0answers
35 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 ...
2
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0answers
23 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
votes
1answer
74 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
79 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
votes
2answers
176 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
53 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
votes
1answer
50 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
votes
1answer
99 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
34 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
67 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 ...
5
votes
1answer
164 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
34 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[ ...
2
votes
1answer
30 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 ...
3
votes
1answer
178 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 ...