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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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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103 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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38 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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25 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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58 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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38 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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40 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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61 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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58 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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37 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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38 views

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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136 views

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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295 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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60 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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28 views

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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203 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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1answer
55 views

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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137 views

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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52 views

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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95 views

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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98 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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234 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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1answer
84 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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228 views

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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184 views

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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143 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, ...
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68 views

Detect bias in subset of Bernoulli processes

I'm looking for advice on the best method to use to answer this question. General scenario: We have multiple testing machines A,B,C,D etc. each tests a identical randomly selected part and provides ...
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368 views

Using continuity correction for normal approximation or not?

Below is a question on a recent actuarial exam, Exam 3L of the CAS. I didn't know whether or not to use the continuity correction when using the normal approximation to do hypothesis testing ...
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Find the expected value of $(\bar X_n-p)^3$

Let $X_1,X_2,\dots, X_n$ be a random sample from a Bernoulli distribution with parameter $p$. Let $\bar X_n$ be the sample average given by $\bar X_n=\frac{1}{n} (X_1+X_2+\dots+ X_n)$). Find the ...
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Why is it futile to use the deviance as a goodness-of-fit measure for Bernoulli data?

In Ordinal Data Modelling by Johson & Albert, page 102-103: For Bernoulli observations [...] the asymptotic chi-squared distribution of the deviance statistic may not pertain. Indeed, for ...
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The probability of getting one variation of consecutive outcomes in Bernoulli trials before another

Consider tossing a fair coin and writing down the outcomes in a sequence of symbols. "H" stands for head and "T" for tail. Let A , B and C be the words HTHH , HHTH and THHH respectively. What is ...
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Using known correlations to select a sample that maximizes model prediction

I want to model the outcome of a binary dependent variable D over a large universe [many records, 100s of variables]. Given a fixed sample size, I want to maximize the accuracy of the model over the ...
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1answer
214 views

exponential density & bernoulli distribution

I had asked a question on Maximum Likelihood earlier. Now I have two questions that are related to this question which had: Let $x$ have an exponential density $p(x|\theta) = \theta e^{-\theta x} ...
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Empirical distribution alternative

BOUNTY: The full bounty will be awarded to someone who provides a reference to any published paper which uses or mentions the estimator $\tilde{F}$ below. Motivation: This section is probably not ...
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211 views

How does the confidence interval for the parameter p of a bernoulli trial vary with the value of p itself?

This is a very basic question (I'm currently studying undergrad level statistics), but I was hoping for some clarification regarding an assertion I read in a newspaper article earlier today. The ...
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130 views

Flipping random coins from a bag - equivalent to a single coin?

My first and I think naive question here. I am trying to model a certain business, and the simplest model I am willing to test is: 1. there is a bag of differently biased coins. 2. every step, a ...
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222 views

Compute a confidence interval for Bernoulli distribution

Given a Bernoulli distribution with a success probability of p = 0.02. Let's say we have N=3000 samples. How can I compute a confidence intervals for the expected number of successes (e.g., with 5% ...
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Dependent Bernoulli trials

The probability of a sequence of n independent Bernoulli trials can be easily expressed as $$p(x_1,...,x_n|p_1,...,p_n)=\prod_{i=1}^np_i^{x_i}(1-p_i)^{1-x_i}$$ but what if the trials are not ...
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What do I need to know about Bernoulli distributions to build a Naive Bayesian classifier?

I'm building a naive bayesian classifier for a binary classification. Right now I have an estimator for Bernoulli distributions, and real distributions (using a kernel mixture distribution). I can ...
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151 views

Wilson's confidence interval for standard deviation?

I have a a set of films, and for each films, a set of reviews - varying between 1 review and several hundred reviews for each film. Each review has a star rating from 1 to 5. I am using Wilson's ...
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358 views

How to show operations on two random variables (each Bernoulli) are dependent but not correlated?

I was looking at the following question from "One Thousand Exercises in Probability" by Grimmett, page 25, question 16 (not homework just self-study): Let $X$ and $Y$ be independent Bernoulli ...
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176 views

Testing median, and distribution of Binomial

Given the median is $m>0$ We have $n$ random variables: $Y_1,Y_2,..,Y_n$ If we observe a value above $m$ then $Y_i=1$ if we observe a value below $m$ then $Y_i=0$ The probability of getting an ...
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K successes in Bernoulli trials, or George Lucas movie experiment

I'm reading "The Drunkard's Walk" now and cannot understand one story from it. Here it goes: Imagine that George Lucas makes a new Star Wars film and in one test market decides to perform a crazy ...