# Questions tagged [probability-theory]

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

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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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-...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n$ ...
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 ...
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 $p$ ...