# Questions tagged [probability-theory]

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

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### Assumption of a generation of the dataset by a probability distribution

Consider the following paragraph from the deeplearningbook The training and test data are generated by a probability distribution over datasets called the data-generating process. We typically ...
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### The expectation of the total number of pairs of keys in a hash table that collide using universal hashing

I am reading CLRS relating to perfect hashing. When computing the $$\mathbb{E}[\sum_{j=0}^{m-1}{n_j\choose{2}}]$$ where $m$ is the number of slots in the hash table, and $n_j$ is the number of keys ...
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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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### How do we prove the time complexity of this simple problem in probabilistic inference on a Bayesian network?

Suppose we have a simple Bayesian network with two rows of nodes: $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_n$. Each node $x_k$ takes a state of either 0 or 1 with equal probability. Each ...
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### 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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### 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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### 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($T, i$) ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $d$-...