# Questions tagged [probability-theory]

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

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### How would you prove a family of functions is universal?

A set of functions from a universe U of keys to n buckets is universal if for every pair of keys in U, say x and y, such that x != y, the probability of h(x) = h(y) is less than or equal to 1/n, for a ...
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### “Learning occupancy grid maps with forward sensor models, S.Thurn,2008” a formula that is difficult to overcome

I am reviewing the document" Learning occupancy grid maps with forward sensor models, S.Thrun, 2008" (http://faculty.iiit.ac.in/~mkrishna/ThrunOccGrid.pdf) i am really confused about formula (26), ...
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### How can Karger's algorithm (and other randomized algorithms) be used in practice?

Suppose I am given the following problem (the source is here): Disconnect two nodes in a graph by removing minimum number of edges. I would apply Karger's min-cut algorithm. But how can I ...
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### Box labelling game

I have a box of stickers. It contains $n$ stickers. Each sticker is labelled with a different number from $\mathbb{Z}$. I have infinite supply of boxes. Box labelling game: I pick a random sticker ...
212 views

### EM-algorithm for categorical hidden variables

I have the following model: Let's say two indepentent weighted six-sided dice $X$ and $Y$ with unknown probabilities (i.e. probability of 1 is unkown etc) and they have not necessarily the same ...
201 views

### Random Walk on the Integer Line

Suppose we are doing a random walk on the infinite integer line and that we take $2n$ total steps. At every step of this walk, the position of the walker is an integer point on this line. For the next ...
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### Reading CLRS analysis of hashing with chaining

I'm currently reading an analysis hashing with chaining, and it goes over two examples: In the first, the search is unsuccessful; no element in the table has key k. In the second, the search ...
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### Expected distance between two random nodes in weighted tree [duplicate]

Given an undirected connected graph with N nodes and N-1 edges. Out of N nodes, 1...k nodes are selected, out of which 2 nodes are selected randomly (uniformly). How do I calculate the expected ...
432 views

### The throughput of the ALOHA protocol if the Binomial distribution was used

In all the examples of ALOHA I've seen, the Poisson distribution is used. Theoretically, how could the throughput be calculated if a Binomial distribution was used instead? For example, in the case ...
186 views

### Deep DFS traverse on graph

According to chess rules, the (undirected) graph generated from the Knight move on a keypad is the following:  \begin{array}{ccccc} 3 & - & 4 & - & 9 \\ | & & | & & |...
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### Classification training data, but regression prediction

Suppose I'm performing machine learning on a simple dataset, and have a bunch of training data of the form: ...
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### Universal lower semicomputable semimeasure and Coding Theorem

I'm following Li and Vitanyi's book "An introduction to Kolmogorov complexity and its applications" 3ed. I'll rewrite here the definitions I need for my question. The authors define the reference ...
203 views

### Clock solitaire game and principle of deferred decision

I have been reading the randomized algorithm book by Rajeev Motwani and Prabhakar Raghavan. In section 3.5 they have introduced principle of deferred decision which is a different probability space. ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 $[n]$ to ...
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### 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 ...
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### 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 1/...
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### 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 ...
363 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 ...
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### 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 ...
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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 ...
680 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 ...
101 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 ...
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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 ...
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. ...