Questions tagged [probability-theory]

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

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21
votes
3answers
3k views

Is rejection sampling the only way to get a truly uniform distribution of random numbers?

Suppose that we have a random generator that outputs numbers in the range $[0..R-1]$ with uniform distribution and we need to generate random numbers in the range $[0..N-1]$ with uniform distribution. ...
26
votes
9answers
17k views

Generating uniformly distributed random numbers using a coin

You have one coin. You may flip it as many times as you want. You want to generate a random number $r$ such that $a \leq r < b$ where $r,a,b\in \mathbb{Z}^+$. Distribution of the numbers should ...
21
votes
9answers
16k views

How to simulate a die given a fair coin

Suppose that you're given a fair coin and you would like to simulate the probability distribution of repeatedly flipping a fair (six-sided) die. My initial idea is that we need to choose appropriate ...
18
votes
1answer
8k views

Applying Expectation Maximization to coin toss examples

I've been self-studying the Expectation Maximization lately, and grabbed myself some simple examples in the process: From here: There are three coins $c_0$, $c_1$ and $c_2$ with $p_0$, $p_1$ and $p_2$...
14
votes
1answer
7k views

Randomized Selection

The randomized selection algorithm is the following: Input: An array $A$ of $n$ (distinct, for simplicity) numbers and a number $k\in [n]$ Output: The the "rank $k$ element" of $A$ (i.e., the one in ...
5
votes
1answer
391 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: ...
5
votes
1answer
324 views

Possible to construct a probabilistic halting problem solver?

I'm a CS undergrad so my math/CS knowledge is not that deep so please correct me if my premise is flawed or I have made some incorrect assumptions. So I was thinking, much in the way that some ...
2
votes
2answers
202 views

Measures and probability in formal language theory

I am looking for references for the following problem: I have a very special class of regular languages and my aim is to express (and to justify my conjecture) that this class itself is very small in ...
7
votes
1answer
170 views

Prove fingerprinting

Let $a \neq b$ be two integers from the interval $[1, 2^n].$ Let $p$ be a random prime with $ 1 \le p \le n^c.$ Prove that $$\text{Pr}_{p \in \mathsf{Primes}}\{a \equiv b \pmod{p}\} \le c \ln(n)/(n^{...
5
votes
3answers
200 views

Unbiasing of sequences

There is the well-known method of unbiasing of bit sequences due to von Neumann. Are there similar schemes applicable to other sequences, e.g. the result of throwing a normal die?
2
votes
2answers
95 views

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($T, i$) ...
1
vote
1answer
132 views

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($T, i$) ...
23
votes
5answers
8k views

How to approach Vertical Sticks challenge

This problem is taken from interviewstreet.com We are given an array of integers $Y=\{y_1,...,y_n\}$ that represents $n$ line segments such that endpoints of segment $i$ are $(i, 0)$ and $(i, y_i)$. ...
5
votes
1answer
1k 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 ...
6
votes
4answers
3k views

What are the uses of Markov Chains in CS? [closed]

We all know that Markov Chains can be used for generating real-looking text (or real-sounding music). I've also heard that Markov Chains has some applications in the image processing, is that true? ...
5
votes
1answer
1k views

Reservoir sampling algorithm probability

I'm reading about the reservoir sampling technique called Algorithm R. The idea is we can take a sample of size $n$ from a population of size $N$ even when $N$ is unknown/too expensive to retrieve in ...
2
votes
3answers
428 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 ...
4
votes
2answers
622 views

Online generation of uniform samples

A source provides a stream of items $x_1, x_2,\dots$ . At each step $n$ we want to save a random sample $S_n \subseteq \{ (x_i, i)|1 \le i \le n\}$ of size $k$, i.e. $S_n$ should be a uniformly chosen ...
3
votes
1answer
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 ...
2
votes
1answer
62 views

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 ...
6
votes
1answer
711 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<\...
4
votes
1answer
171 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}...
3
votes
1answer
196 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}}, ...
3
votes
1answer
105 views

Finding the dichotomy that maximizes information gain for a classifier?

Suppose that $\Omega$ is a finite probability space,with measure $P$ (we can take $P$ uniform). Let $C \in \{\pm 1 \}$ be a random variable on $\Omega$, the classifier. Let $$H(C) = H(C, \Omega, P) = ...
2
votes
2answers
153 views

The optimal way to reverse engineer a binary classification problem

I am not sure if this question is more suitable for CS, theoretical CS or math so feel free to improve the description and migrate it. In a scenario very similar to popular binary classification ...
2
votes
1answer
70 views

Randomized BST height analysis : How $Z_{n,i}$ and $Y_{k-1}$ are independent?

I am referring to this video https://www.youtube.com/watch?v=vgELyZ9LXX4 at 1:08:39 . $n$ : number of nodes in the tree $Z_{n,k}$ : Indicator random variable that activates when rank of the root ...
0
votes
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 ...
-3
votes
1answer
784 views

m-element random sample being equally likely …(CLRS 5.3-7)? [closed]

I am trying to understand the following solution to CLRS 5.3-7: http://clrs.skanev.com/05/03/07.html Question description is on the page. I understood the part where m-element subset is constructed ...