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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Bernoulli maximum likelihood

Suppose $X_1, X_2, \dots, X_n$ are iid Bernoulli(p) random variables. How do you find the restricted maximum likelihood for p where $0<p<0.5$? My work so far: Write out the likelihood: ...
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can you analytically solve this bayesian hierarchical model - bernoulli trials

Is it possible to analytically solve (i.e., use a conjugate prior) the hierarchical model shown in the image below to obtain the posterior distribution. The data are composed of bernouli trials ...
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probability of a success?

This is related to a simple greenhouse effect simulation. A photon of infrared radiation starts at the surface of the planet. On its path into space, it can have in its path 1, 2, or 3 molecules of ...
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Maximum likelihood estimation of p in a Binomial sample

Assuming I need to find the ML estimator for p, p being the chance of success in a Binomial experiment $Bin(N,p)$, I would expect my density function to be: $$ f(y) = {{N}\choose{y}} p^y(1-p)^{N-y} ...
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Variance of a product of Bernoulli with another distribution

This is probably a stupid question, so my apologies if this is too simple. I have a distribution X, now I play the following game: I toss a coin, if it falls on a head, I get nothing, if it falls on ...
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Convergence of standardized means of a Bernoulli variable / CLT

The Question Consider a binary random variable X that satisfies: $Pr(X = 0) = \theta \ \ \ $ and $Pr(X = 1) = 1−\theta $ for $\theta \in (0, 1)$ an unknown parameter. Suppose an i.i.d. sample of size ...
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Hypothesis Test on Contest, a problems?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
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Dependent Bernoulli trials confidence interval

I would like to know if there is a way to build a confidence interval, for a random variable which has a Bernoulli distribution, based on its history. I mean if the order of its states is 11100 (i.e. ...
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39 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 ...
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112 views

How to compute the PDF of a sum of bernoulli and normal variables analytically?

Can convolution be applied to get a closed form expression for $Z = X + N$ where $X$ is a Bernoulli random variable and $N$ is a zero mean normal random variable independent of $X$?
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168 views

Logistic Regression: Bernoulli vs. Binomial Response Variables

I want to perform logistic regression with the following binomial response and with $X_1$ and $X_2$ as my predictors. I can present the same data as Bernoulli responses in the following format. ...
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51 views

Zero-inflated Poisson and Gibbs sampling, proofs and sampling

I am trying to figure out zip-inflated Poisson (ZIP) model. In this model, random data $X_1, .., X_n$ are of the form $X_i=R_iY_i$, where the $Y_i$'s have Poisson distribution ($\lambda$) and the ...
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Is a random variable Bernoulli? Is a proof available?

Suppose a die is tossed twelve times and each outcome is represented by a random variable $X_{i}$. Further define $Y_{i}$ for $i=2,...,12$ to take the value $1$ if $X_i=X_{i-1}$ and $0$ otherwise. ...
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1answer
37 views

Sampling Twice and Posteriors

I have a random variable with some unknown distribution with support over $[0, 1]$. Every turn, I sample a $p_t$ from this distribution. However, I am unable to observe $p_t$ directly. Instead I ...
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16 views

Which distribution should I better use to predict the response in {0,20} applying GBM? [duplicate]

I want to predict the response that is in {0,20}. I am using GBM to make the prediction. ...
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28 views

Maximum Entropy with no index

This is a simpler problem than trying to solve, but have a feeling once get the methodology I can apply it to the harder problem. Let $ H(p)= -q \ln(q) - p \ln(p) $ be the entropy of the Bernoulli ...
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Estimating conditional probability of bernoulli data

Assume I have $i=1,\dots,N$ fathers, each with $j=1,\dots,n_i>0$ sons. Now there is a binary event $A_{i,j}$ with outcomes 1 and 0 and the respective probabilities $p$ and $1-p$. Now I want to ...
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44 views

Accounting for non normality

I have a random variable, supposedly from a Bernoulli distribution, and I want to do some tests on its mean. I'd like to assume that the sample mean is normally distributed, but I'm not sure how ...
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38 views

Estimating the true distributions from a sample of distributions

I am having a hard time formulating the following problem. Consider a company that runs a survey across several cities in the US to estimate the percentage of right-handed people and left-handed ...
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286 views

Expected number of tosses till first head comes up

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time. What is the expected number of tosses that will be required? What is the expected number of tails that will ...
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16 views

Constructing hypothesis tests and error statistics for a bernoulli distribution

I'm trying to use a bernoulli distribution (which seems simpler than a normal distribution) to get a grip on the basic construction of hypothesis tests. So, we have a coin. Its probability of ...
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Linear combination of discrete variables $T_i$ with $P(T_i=1)=P(T_i=-1)=1/2$

Let $T_1,...,T_n$ be iid with a Rademacher distribution; i.e., $P(T_i=1)=P(T_i=-1)=1/2$; and let $w = (w_1,...,w_n) \in \mathbb{R}^n$ without further constraints on $w$. Is there a way to compute ...
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131 views

Sample size needed to estimate probability of “success” in Bernoulli trial

Suppose a game offers an event which upon completion, either gives a reward, or gives nothing. The exact mechanism for determining whether the reward is given is unknown, but I assume a random number ...
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Fisher information matrix determinant for an overparameterized model

Consider a Bernoulli random variable $X\in\{0,1\}$ with parameter $\theta$ (probability of success). The likelihood function and Fisher information (a $1 \times 1$ matrix) are: $$ \begin{align} ...
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93 views

When to use a normal approximation of a Bernoulli distribution

What is a practical example where one would want to use a normal approximation of a binomial distribution over using properties of the binomial distribution itself? I.e if I already know that the ...
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131 views

Text Classification using TfIdf and Bernoulli NB

So, as I am reading about Bernoulli distribution and text classification, I want to understand how Bernoulli uses TfIdf features? Since TfIdf values are within [0-1) but Multivariate Bernoulli assumes ...
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44 views

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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72 views

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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76 views

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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27 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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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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146 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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1answer
127 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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21 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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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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94 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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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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191 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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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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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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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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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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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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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? ...