Questions tagged [bernoulli-process]

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Does the hypergeometric distribution follow a Bernoulli process?

There is a comment in an online resource that I need help understanding: In a Bernoulli process, given that there are M successes among N trials, the number X of successes among the first n trials ...
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Converting likelihood ratios hypothesis test from equality to inequality

I have the following LR test for proportions (to anyone familiar to applied statistics in finance, this is known as the Christoffersen's unconditional coverage test). We would like to know if $\pi$ is ...
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For n Bernoulli trials whose outcomes are 1 or -1, what is the PMF of the sum of n trails being r

Let $x_1, x_2, ...$ be iid with probability $p$ of being 1 and $1-p$ of being -1. Let $S(n) = \sum_{i=1}^n x_i$ and $S(0)=0$ Find $P(S(n)=r)$. My attempt: At first this looks like a normal Bernoulli ...
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Bernoulli process and two exponentials

Suppose that a very long Bernoulli process gives a sequence with possible values: $A$ with probability $p$, and $B$ with probability $1-p$. The expected fraction of contiguous sequences of length $k$ ...
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2 votes
1 answer
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Rendering Bernoulli sampling by the binomial distribution

I want to characterize the simple algorithm for selecting, say, 20% of a discrete set when the total size of the set is not known: ...
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How can I calculate the probability of K zero-crossings occurs in Bernoulli process at given length L?

In the Bernoulli process B[k] = 1(prob. p), -1(prob. (1-p)), the probability of B[n] !=B[n-1] will be (B[n] = 0 && B[n] = 1) + (B[n] = 1 && B[n-1] = 0) = 2p(1-p), so the probability ...
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Number of Samples for Accurate Measure of Heavily Weighted Coin

I see articles on how to prove the "fairness" of a coin by flipping it N times. Using the estimator of true probability p=h/(h+t) along with a target max error and confidence interval, you ...
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How many samples to state $P(A < B) \ge 95\%, A, B \sim \text{Bernoulli}$? [closed]

Suppose I have two series of samples: $$A \sim Be(p)$$ $$B \sim Be(q)$$ How may samples samples do I need to state that: $$P(A<B) \ge 95\%$$
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PDF of estimated Bernoulli parameter

This is my first question on this part of the stack exchange, so please bear with me and correct me if I am missing something obvious. Statistics is not my main field of expertise. Background I wish ...
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Standard Deviation of Bernoulli trials / Bernoulli process with different probabilities

If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure, what is the standard deviation of the result? How is this calculated?
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confidence intervals for the Poisson process ($\lambda$) sampled with uncertainty

Say, I have a Poisson process which was measured $N$ times, and each measurement produced $k_i$ value. Also, $k_i$ are events that I have to detect and my detection probability is $p$. In fact, I ...
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3 votes
1 answer
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Flat "geometric distribution" by varying the probability of the Bernoulli trail

In a simulation I am working on, each day (time step) there is a chance that a condition changes (at which point it is stuck in the changed condition). Setting this probability to a fixed value (say 5%...
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2 votes
1 answer
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How to calculate uncertainty of probability derived from random samples?

I'm running many simulations based on random samples, each of which produces a True or False result. The goal is to calculate the probability that the result will be True, which I can easily ...
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Bernoulli Cusum Chart?

Can I use the R package named 'cusum' to make bernoulli cusum charts or should I use another package? I've been asked to explore using Bernoulli cusum chart for trending a parameter that's usually 0 ...
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1 answer
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How can Bayes avoid Cromwell?

I'm studying widgets and their failures. Generally a widget will run for many years without trouble, but 1-2% of widgets will fail in a given year. I have a table which lists widget manufacturers (A, ...
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Bayesian hierarchical coin flip model

My question is: what is the marginal probability $P(x_1, x_2, \dots, x_n | y_1, y_2, \dots, y_n, \alpha, \beta)$ or $P(X|Y, \alpha, \beta)$? in the following model: $\phi \sim \text{Beta}(\alpha, \...
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3 answers
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How to calculate a confidence interval for a series of Bernoulli Trials?

I have to test if a event have a p probability of happening. I can run this event as much times I like (given it can be run by a computer). So I was searching a way ...
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How can I use the distribution of run lengths to test if a sequence is generated from flips of a fair coin?

I have a very long sequence (in the tens of thousands) of binary outcomes from some data-generating process. I believe that these outcomes are iid Bernoulli trials with p = 0.5, equivalent to flipping ...
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Question related to equation in paper involving bernoulli process

So I'm reading a paper about optimization of design of structures. The author defines H(p) as the function that returns the cost of failure given decision variable <...
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Detecting change in p of a Bernoulli process

A machine outputs either a 0 or a 1 each second. We denote this output at time $t$ as $b_t$. The probability that it outputs 1 is $p_t$ at time $t$. How do we go about studying the change in $p_t$ in $...
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How to determine minimum sample size for Bernoulli trials at a given confidence level?

I want to determine whether the true bounce rate of an email campaign (an email sent to many recipients) is <20%, at the 99% confidence level. I can send one "test batch" of randomly selected ...
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7 votes
0 answers
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Is average stopping time a continuous function of Bernoulli parameter?

Consider an infinite sequence $X = (X_i)_{i \in \mathbb N}$ of i.i.d Bernoulli random variables with (unknown) parameter $p \in (0,1)$, and let $N$ be a stopping time on $X$. Is it always the case ...
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What is a name of this Bernoulli-like process with dependent trials?

The process is defined similarly to the Bernoulli process composed of $n$ Bernoulli trials. The difference is that the trials are dependent, that is: $$ P(X_i = 1 | X_1, ..., X_{i-1}) = \frac{m -\...
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Independent Bernoulli trials vs markov chain

Original Question Suppose we have a sequence of Bernoulli trials $X_1, X_2, \cdots X_T$ which are ordered in time and may or may not be independent. I am interested in understanding the probability of ...
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Transforming Categorical (a.k.a. Nominal) data to Bernoulli Variables

I've been doing some multivariate analysis for a dataset that contains, for the most part, categorical data. For example, I have two which are: gender (M or F) state (A, B or C) and each observation ...
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1 vote
2 answers
356 views

What type of statistical test should be used to compare successes of two treatments?

Say I have two treatments, "hot" and "cold". I care about a certain outcome, say "success". I have data on the number of successes out of, say 100 trials in the "hot" treatment. Similarly, I have data ...
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Write a Bernoulli distributed random variable using uniforms

Consider the real-valued random variables $\{X_{ij}\}_{\forall (i,j)\in \mathbb{N}^2, i<j}$ and the real-valued random variables $(\xi_0, \{\xi_i\}_{\forall i \in \mathbb{N}})$, where $\mathbb{N}$ ...
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3 votes
2 answers
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Success of Bernoulli trials with different probabilities and without replacement

Assuming $n$ independent Bernoulli trials with different probabilities, the Poisson binomial distribution is the discrete probability distribution that describes the number of $X$ successes. A ...
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2 votes
1 answer
158 views

Bernoulli parameter in the following partitioning schemes

Suppose I have a Bernoulli process: $2n$ Bernoulli observations whose locations are distributed over an interval, say $(0,1]$. Conditional on the locations, $x_k\stackrel{iid}{\sim}Bern(\theta)$, $k=1,...
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6 votes
1 answer
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Number of Bernoulli trials to first success, with changing $p$

[A version of this question was previously posted by another user, but the OP deleted rather than edit the question into a more suitable form for routine textbook work. I am reposting in the hope of ...
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Maximum likelihood estimation of a Poisson binomial distribution

According to Wikipedia, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed In ...
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5 votes
1 answer
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Age and residual life time of the Poisson process

Original Question Let $N(t)$ be a Poisson process with intensity $\lambda$. Let $T_1<T_2<...$ be the occurrence times. Let $T_0=0$. For any $t>0$, define the $age$ random variable to be $...
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Training set for Bernoulli Process: retain number of "1" examples in proportion to process?

Given a Bernoulli Process, should my training set have a number of "1" examples in proportion to the process? For example, a Bernoulli Process is "1" 10% of the time and "0" otherwise. In a training ...
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Combinations of Bernoulli Trials

I recently asked another question, which I have linked here: Combining Binomial Random Variables. I wanted to add onto that question, so I am asking in a different thread. Brief recap of previous ...
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Understanding Bernoulli and logit function

currently I am reading a paper and trying too implement what is in the paper by myself. I plan to implement using R. I'm stuck at below part: I understand the X and Z but I'm not familiar with ...
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Convergence of a sequence of Binomial variable with changing probability

Consider a $t\in(0,1)$. Consider, for $\Delta>0$ the random variable $X_t^{(\Delta)}$ defined as $$ \mathbb{P}[X_t^{(\Delta)}=1]=\left(1-\lambda\,\Delta\right)^{\left\lfloor t/\Delta\right\...
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1 vote
1 answer
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How many Bernoulli trials to have N successes in series with restarts?

Bernoulli trials are sequences of 0 and 1. What is the average length, the number of trials, that you need to perform until you reach n ones in sequence? I tried to make it with a 2D recursion Here ...
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0 votes
1 answer
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What is the value of π in this experiment(Bernoulli)?

Experiment: You roll a fair 6-sided die 5 times. Define the random variable x = number of times you rolled an even number. The probability of exactly X successes in n trials for a Bernoulli process ...
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5 votes
1 answer
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Number of trials necessary to demonstrate Bernoulli process doesn't have mean p

I have a Bernoulli process that purportedly has mean $x$ but I hypothesize that the process actually has mean $q$. How many trials are necessary to demonstrate (to some confidence $p$) that the actual ...
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Distribution of number of Bernoulli trials before some large number of sucess [duplicate]

We repeatedly make an experiment where we count the number $n$ of Bernoulli trials of known probability $p$, until some number of successes $s$ is reached. I'm willing to restrict to $p<0.01$ and $...
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60 votes
13 answers
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Does 10 heads in a row increase the chance of the next toss being a tail?

I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of ...
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2 votes
1 answer
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Bit Error Rate - Multiple Bernoulli trials with different probabilities

If you transmit a sequence of bits over some line, errors may creep in. For instance, if the Bit Error Rate of this line is 1%, on average, every 1 out of 100 bits will flip to the opposite value. ...
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Bounds on binary event estimation [duplicate]

I would like to paint an objective picture of some binary outcome. Now I have data like this: 1085x yes, 1704x no. The percentage of the positive outcome is 40.72%, but I want to give an estimation ...
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2 votes
1 answer
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Sum of Beta-Bernoulli variables

Assume you have $x_i \sim \operatorname{Bernoulli}(p_i)$ with $p_i \sim \operatorname{Beta}(\alpha,\beta)$. I am exploring $Z=X_1+ \dots +X_n$ According to this page, it is $Z \sim \operatorname{...
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2 votes
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Subsample of a random sample?

I am stuck with a very simple question, but I don't really understand sampling, so please help me. Assume that I perform Bernoulli sampling with parameter $q$ on data D, and obtain sample S1. Then on ...
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2 votes
1 answer
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How does this trick for estimating bit error rate break down?

Imagine you are receiving a message over and over via a lossy data path. The path causes bit errors but does not affect the length of the message (or shift any bits). You don't know the actual ...
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