The Bernoulli distribution is a discrete distribution parametrized by a single "success" probability. It is a special case of the binomial distribution.

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Independent events in sequence

If I flip a fair coin twice, the probability of at least one Head is 0.75 (HH, HT, TH and TT). I flip the coin once and it lands Tails. In order to get at least one Head, the probability of the ...
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How to prove Bernoulli distribution belongs to the exponential family

According to a book, a distribution belongs to the exponential family if it can be written in the form of I wrote the Bernoulli distribution as $\exp\Big(y \log\,[{\mu}/{(1-\mu)}] + ...
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Sequential testing for a Bernoulli variable

Suppose $T$ is a Bernoulli random variable, such that $T = 1$ with probability $s$, $s$ is not known. I am interested in checking hypothesis $H_1$ that $s \neq 1/2$ ($H_0$ is that $s = 1/2$). One way ...
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Logistic Regression - Error Term and its Distribution

On whether an error term exists in logistic regression (and its assumed distribution), I have read in various places that: no error term exists the error term has a binomial distribution (in ...
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19 views

Computing mixture of Binomial distributions

I'm trying to model a simple Bayes net, with $n$ samples based on a (unobservable) Bernoulli parameter, representing a true state of the world. Let $T$ be a Bernoulli random variable, with ...
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Is there a test of pair-wise independence for a sequence of non-identical Bernoulli random variables?

Given a sequence of $n$ instances of Bernoulli random variables $x_1,\ldots,x_n$, I am interested in testing whether $x_i$ is independent of $x_{i-1}$ for $i=2,\ldots,n$ (i.e. ...
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30 views

Testing for samples being drawn from identical Bernoulli r.v.'s

Suppose that I have a sequence of size $n$: $x_1,\ldots,x_n$, $x_i\in\{0,1\}$. My null hypothesis is that all $n$ members of the sequence are drawn independently from an identical Bernoulli ...
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52 views

Mean and Variance for a sum of independent weighted bernoulli random variables with different probabilities of success

Suppose $Z_i$ are independent Bernoulli random variables with differing probabilities $P_i$. Also suppose weights $W_i$ are positive and constant. Can you tell me the mean and variance for the ...
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117 views

maximum-likelihood of a sequence of events described by a Bernoulli distribution

I am having quite some troubles with the following homework: In a city it's measured for the whole year whether it rained or not. A distribution $\textrm{Bernoulli}(r_t|\rho)$ characterizes the ...
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17 views

Change detection in a Bernoulli sequence

Suppose $X_1, X_2, \ldots, X_n$ is a sequence of independent Bernoulli random variables with (say) $$X_i \sim \begin{cases} \mathrm{Ber}(0.1) & 1 \leq i \leq an \\ \mathrm{Ber}(0.2) & an+1 ...
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432 views

Test if two binomial distributions are statistically different from each other

I have three groups of data, each with a binomial distribution (i.e. each group has elements that are either success or failure). I do not have a predicted probability of success, but instead can ...
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51 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 ...
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Is there a name for this process/ distribution?

Does the equation below have a name, or is it similar to some other well-known process/ equation? Equation of interest: $$S_c = S_{c-1} + S_{c-1}\omega_c\delta_c$$ $\delta\sim\mathcal{N}(0,1)$ is a ...
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Probability function of number of fails to first success

I quote the question: In some cultures, it is important to have at least one son. The plan is to have children until one son is born. Find the PDF of the number of daughters in a family. The ...
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67 views

Simple scheme to sample from the Bernoulli distribution

I was implementing a simple scheme of Bernoulli distribution sampler. $ X \sim B(p) $. I have a function that generates a uniform random number $r \in (0,1)$. Then, I set $ X = 1 $ if $p > r $, ...
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Bayesian hypothesis test

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
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127 views

Hypothesis test for Bernoulli

Let $X_{1},..,X_{36}$ be a sample from a Bernoulli distribution with parameter $p$. The sample proportion is $\frac{1}{3}$. Consider a normal approximation test $H_{0}:p=0.5$ vs $H_{1}:p\neq 0.5$, ...
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252 views

How can I determine statistical significance in an A/B test in which the KPI is dependent upon two variables - one bernouli and one continuous?

In my work in online marketing, we frequently run A/B or multivariate web page tests. The Key Performance Indicator for these tests is overall Revenue. The treatment that nets the most revenue, either ...
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259 views

Sum of Bernoulli variables with different success probabilities [duplicate]

Let $x_i$ be independent Bernoulli random variables with success probabilities $p_i$. That is, $x_i=1$ with probability $p_i$ and $x_i=0$ with probability $1-p_i$. Is there a closed expression or an ...
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44 views

Determine if many coins are fair

Imagine I have many coins ($>100$), and for each coin I have the data: $(n_{\rm heads}, n_{\rm tails})$ where $\sum(n_{\rm heads}, n_{\rm tails})$ is decreasing for each next coin. Any one of the ...
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Comparing Bernoulli means across subpopulations in which the number of observed successes may be zero

I've got a binary characteristic and a population $S$ with size $n$ and $P[X] = p$ such that $p$ may be small and $n$ is extremely large. Within this population are subpopulations of various sizes ...
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34 views

tight bound of bernoulli sums with unknown dependency

Consider n random variables $X_1, \ldots, X_n$ all follow same bernoulli distribution of mean $p$. But the dependency of these variables are unknown (i.e., cannot assume that they are independent). ...
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The probability that one bernoulli process has a higher p than another?

I have two data generating processes that are independent Bernoulli processes with probabilities of success $p_A$ and $p_B$. I am taking repeated samples from these two data generating processes, so ...
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162 views

Normal Approximation of the sum of correlated Bernoulli Random Variables

Hi I am looking for a result (if it exists !!!) in the direction of Normal approximation for sum of correlated Bernoulli random variables (edit : with the same parameter $p$) where correlation between ...
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108 views

CLT can be used for weighted sum of different Bernoulli variables?

Suppose $$ z_i \sim Bernoulli (p_i) $$ Can we use CLT for the following weighted sum? $$ S = \sum_i w_i z_i $$ i.e. can $S$ be approximated with a normal distribution? If yes, with which theorem? ...
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126 views

Binom.test error in R: “ formal argument ”p“ matched by multiple actual arguments”

I'm running a binom.test on the data set UCBAdmissions (comes with R) and am stuck on an error message. About the data: ...
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91 views

Multiple Bernoulli and Multinomial Distirbution

It's well known that language can be modeled by Multinomial distribution and Multiple Bernoulli distribution. So far I don't see any advantage of Multiple Bernoulli distribution representation over ...
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211 views

How to get distribution of sum of dependent bernoulli variables

I have $N$ Bernoulli variables, $X_1,...,X_N$ and $X_i\sim B(1, \pi_i)$, $\pi$ is known for each $X_i$, and $Y=X_1+...+X_N$, now I need to get the destribution of $Y$. If $X_i$ and $X_j$ are ...
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49 views

Relationship between Bernoulli and Normal CDF

Is there any relationship between the draw from a Bernoulli with parameter $p$ and the Normal CDF. Specifically is the condition $p>\Phi(x)$, where $x$ is drawn from the standard normal ...
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Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
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Estimating successes while obtaining Bernoulli samples

I have a process which, after fixing the values of some parameters, generates samples from a Bernoulli distribution with unknown $p$. The value of $p$ is typically small, and what I want to do is to ...
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648 views

What does a confidence interval with a negative endpoint mean?

I have 10 iid r.v. with Bernoulli distribution with $X_{i} = 1$ for a positive result. I'm given $\sum_{i=1}^{10} X_i= 1$ and need to find a two-sided 99% confidence interval for $\theta$. So ...
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124 views

Expected value of an indicator function

We have $$ I_i (x) = \begin{cases} 1 \quad x-h<X_i<x+h \\ 0 \quad \text{otherwise} \end{cases} $$ Does that indicator function follow the Bernoulli distribution? If so, the expected value can ...
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Is the mean of n independent Bernoulli random variables normal? [duplicate]

I'm trying to figure out the difference between an A/B split test and an ANOVA test and I came across this article. It suggests that the mean of $n$ independent Bernoulli random variables is normally ...
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Bernoulli Confidence Intervals for p very close to 0

Let's say I have the following observations from many Bernoulli distributions with different p (p1, p2, ..): Observations from Distribution 1: 10 successes, 100,000 trials, p_hat = 0.0001 ...
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224 views

p-value of hypothesis “a home court advantage exists”

The practical POV: I sample 35 matches: 15 turn out as home win, 10 as tie and 10 as home lose. I want to conduct a test for the hypothesis "A home court advantage does exist." with a significance ...
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Using cross correlation to infer dependence, can it be done?

I have a very particular question, I have seen a similar one here, but my knowledge is too limited to make use of it. I will try to explain myself as clearly as possible... Wish me luck! I have a ...
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Are $\mathbb{F}_2$-linear combinations of random variables in an i.i.d. Bernoulli process again an i.i.d. Bernoulli process?

I'm having trouble understanding how certain combinations of random variables can correlate. The problem is as follows: I have a binary $m \times n$ matrix $A$ of full rank (over the finite field ...
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Flipping identifiable coins in batches

I have collected data to estimate a parameter and am now puzzled about how to generate confidence intervals: Setup: 1) We have a bag with $N$ coins. 2) Each coin $i \in N$ has a known probability ...
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Significance test for highly skewed Bernoulli distribution

I am working with two highly skewed Bernoulli distributions where 96-99+% of the samples are in the "false" category, and the rest are in the "true" category (sort of speak). I am looking for a ...
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223 views

Standard Deviation of fixed-odds bet

I'm looking to calculate the standard deviation of a fixed-odds betting proposition. The bet pays 5/6 for a win and you lose your stake for a loss. For a bet of 100, a winner pays 83.33 (and the stake ...
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462 views

Bernoulli random variable parameter estimation

Suppose $\theta$ is the probability that a Bernoulli random variable is one (therefore $1-\theta$ is the probability that it's zero). I have a sequence of $n$ of these i.i.d. Bernoulli random ...
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Estimating the parameters of a normal distribution of probabilities

I know the question sounds weird or perhaps, silly. And my limited knowledge in statistics might have caused ambiguity in presenting the problem. But still, here's the case: Suppose there's a ...
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106 views

How do choices of probabilities and covariance matrices constrain each other for a correlated multivariate Bernoulli random variable?

I have a correlated multivariate Bernoulli random variable $\textbf{X} = (X_1, ..., X_N)$, where the $X_i$ are Bernoulli random variables with parameters $p_i$ and $N \times N$ covariance matrix ...
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How can I sample from a correlated multivariate Bernoulli distribution with known covariances?

I have $N$ Bernoulli random variables $X_1, ..., X_{N}$ with known parameters $p_1, ..., p_{N}$. They are dependent with known covariances. How can I sample from the joint distribution of the $X_i$? ...
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Designing an experiment: Geometric or Bernoulli data

I have some process that succeeds or fails with probability $p$. I can do distinct simulations to estimate $p$: Run $N$ simulations of a single process, record $N$ samples of a ...
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Sum of Products of Rademacher random variables

Let $x_1 \ldots x_a,y_1 \ldots y_b$ be independent random variables taking values $+1$ or $-1$ with probability 0.5 each. Consider the sum $S = \sum_{i,j} x_i\times y_j$. I wish to upper bound the ...
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Should coin flips be modeled as Bernoulli or binomial draws in RJags?

What is the best way to model coin flips as a hierarchical model? Do you say coin draws are a series of draws from Bernoulli trials or as one draw from a binomial distribution? That is something like ...
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Generalization of Chernoff bound to Bernoulli schemes

I have this (I think) simple problem, which I guess is just a generalization of the Chernoff bound to Bernoulli schemes. Assume I have a bowl with N balls with R different colors. The balls are not ...
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173 views

Simple mean reversion measure for binary time series

I am trying to define a simple measure for mean reversion in a stochastic sequence of ones and zeros, which I denote by $x_t$. Yes, a unit root test on the cumulative sum could be a viable choice, ...