Questions tagged [bernoulli-process]

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Use a sequence of bernouli trials to find the possibility of an event happening

I have the data below and We have a game, that has a prize jackpot. The jackpot prize starts at 1,000 credits and every round you play the jackpot increases, until someone wins the jackpot and it ...
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Are paired matches in ranked data independent, if the original, unranked, continuous pairs are independent?

Let $(X, Y)$ be a randomly drawn sample of $n$ paired observations from a bivariate continuous population. It is clear that each pair is independent of the others, both pairwise and mutually. Let $R(X,...
virtuolie's user avatar
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Estimating the probability of success in a sequence of Bernoulli Trials when the probability is changing over time

Say I have a time series X from Bernoulli Trials with outcomes 0 or 1 where X(n) is the $n$th outcome in the time series. The process is driven by some probability of success $\pi$ but this ...
Oscar's user avatar
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3 votes
1 answer
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One statistical test for 4 subsets at a time

Imagine a card player who is participating in a tournament with 4 different versions of one card game. The probabilities of winning by doing random actions in each type of game are the following: 0.61,...
Igor Igor's user avatar
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1 answer
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Logical disjunction OR between two independent random variables

Consider the Bernouilli experiment of tossing a coin $2$ consecutive times, with the probability of getting "heads" of $p=0,8$ The base space can be described as follows $\Omega=\{HH,TT,HT,...
niobium's user avatar
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Binomial Proportions when sampling without replacement not studied?

Question When doing surveys with a population, we obviously don't replace by design. Why is it then that the mathematics and statistics described in textbooks seems to be veering more on the side of ...
user3659451's user avatar
14 votes
1 answer
789 views

DataCamp exercise about distributions

I was studying some statistics in DataCamp and they assigned me this exercise that I can't solve. I tried speaking with people that know more statistics than me and we can't seem to agree in an answer....
Lucas Giraldi's user avatar
2 votes
1 answer
43 views

Length of a bernoulli process to get a fixed amount of tails

I have the following Bernoulli process: Initial state: H = 0, T = 0 I launch a coin where probability of head is p. If I get head, then H = H + 1. If I get tails, then T = T + 1. I repeat the ...
ABu's user avatar
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1 answer
420 views

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 ...
Snehal Patel's user avatar
3 votes
1 answer
90 views

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 $k$ ...
user1420303's user avatar
2 votes
1 answer
112 views

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 ...
Ryan's user avatar
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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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1 vote
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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?
MintDice's user avatar
4 votes
1 answer
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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 ...
Gideon Kogan's user avatar
3 votes
1 answer
233 views

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%...
goryh's user avatar
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2 votes
1 answer
866 views

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 ...
endolith's user avatar
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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 ...
variable's user avatar
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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, ...
Charles's user avatar
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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, \...
James Hay's user avatar
3 votes
3 answers
2k views

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 ...
Bruno Andreetto's user avatar
2 votes
1 answer
130 views

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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1 answer
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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 <...
jpcgandre's user avatar
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2 answers
320 views

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 $...
ztyh's user avatar
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2 votes
1 answer
1k views

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 ...
M. Giangreco's user avatar
7 votes
0 answers
128 views

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 ...
Luis Mendo's user avatar
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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 -\...
abukaj's user avatar
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8 votes
1 answer
992 views

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 ...
Daniel Severo's user avatar
1 vote
2 answers
1k 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 ...
Atticus29's user avatar
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0 answers
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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 $\mathbb{N}$ ...
Star's user avatar
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3 votes
2 answers
3k views

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 ...
user2597079's user avatar
2 votes
1 answer
209 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,...
stats134711's user avatar
6 votes
1 answer
1k views

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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6 votes
1 answer
1k views

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 ...
anonymous's user avatar
5 votes
1 answer
3k views

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 $...
Ye Tian's user avatar
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1 vote
0 answers
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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 ...
nbui's user avatar
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0 answers
265 views

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 ...
wsavran's user avatar
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1 answer
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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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0 answers
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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\...
AlmostSureUser's user avatar
1 vote
1 answer
181 views

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 ...
Valentin Tihomirov's user avatar
0 votes
1 answer
3k views

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 ...
JakeFromStateFarm's user avatar
5 votes
1 answer
154 views

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 ...
Cirdec's user avatar
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1 answer
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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 $...
fgrieu's user avatar
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61 votes
13 answers
56k views

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 ...
user68492's user avatar
  • 611
3 votes
1 answer
489 views

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. ...
LBogaardt's user avatar
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0 answers
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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 ...
Peter Smit's user avatar
4 votes
1 answer
2k views

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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