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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What is the Bernoulli class conditional distribution?

What is the Bernoulli class conditional distribution? I am trying to implement a procedure for computing a naive Bayes classifier for binary features with a Bernoulli class conditional distribution. ...
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Terminology clarification: Discrete distribution == Categorical distribution?

I'm reading "The Indian Buffet Process: An Introduction and Review" by Griffiths and Ghahramani and wanted to confirm my understanding of one of the terms they use. On page 1188, they say that the ...
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Estimate point in metric variable where probability of success of a conditionally Bernoulli distributed variable changes (independent observations)

There is a metric variable X and a conditionally Bernoully distributed variable Y, where the probability of success of Y changes at a threshold x of variable X. The obervations are independent. I want ...
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Bernoulli process of not-fixed probability p?

Here's a question that might be basic, but I want to understand what distribution(s) would result when the underlying Bernoulli trials' probability $p$ varies with time...?? I assume that it's not the ...
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Building compound distribution of Bernoulli and geometric distribution

Let $ X \sim \text{Bernoulli} (N, p)$ where $N \sim \text{Geometric }(\theta)$ Then how to calculate P(X=x), E(X) and V(X)? If $N \sim \text{Geometric }(\theta)$ then $f(n) = \theta ...
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Success rate estimate and error from a sample of a Bernoulli distribution?

I have a sample of size $n$ of independent trials extracted from a Bernoulli distribution with unknown success rate $\theta$. Given this sample, how can I estimate the success rate $\theta$ along with ...
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Expectation of Bernoulli r.v. of arbitrary support $k\in\{a,b\}$?

How does one adapt the properties of Bernoulli variable to an arbitrary support $k\in\{a,b\}$, when the Bernoulli is typically defined for $k\in\{0,1\}$? I'm specifically interested in ...
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Classifying the F7 (First 7 days Activity) Metric

In social networks we have a commonly used metric called F7. It is the number of days a user logged in during their fist week since sign up. This means it has a range of [1,7] since you must at least ...
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Q-Q Plot for Simulation Study with Bernoulli Data Issue

I'm currently trying to carry out a simulation study for Bernoulli Data to show that the sample proportion, ˆp, is also approximately normally distributed when the sample size is large. From an ...
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Standard Deviation vs. Standard Error of Bernoulli Trials

There is a related question and clear answer here: http://stats.stackexchange.com/a/11542/22088 But my question is slightly different. I posted a comment but the answerer has not been on Stackexchange ...
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Expected value of the ratio of shifted Bernoulli variables

Have two independent Bernoulli variables $X,Y$ I'm trying to calculate $$\mathbb{E}\bigg[\frac{X+1}{Y+1}\bigg]$$ Given $X,Y$ are independent, this can be split into: ...
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Mean of product of independent Bernoulli random variables

I am having difficulty with this. The product of independent Bernoulli random variables is Bernoulli. I am unsure as to how to go about solving this problem.
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Std. error of accuracy on 3-class problem

I want to find the std. error of accuracy on a 3-class problem, where the following is true. Number of observations is N=40000, number of classes is ...
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1answer
61 views

How to derive the likelihood function for binomial distribution for parameter estimation?

According to Miller and Freund's Probability and Statistics for Engineers, 8ed (pp.217-218), the likelihood function to be maximised for binomial distribution (Bernoulli trials) is given as $L(p) = ...
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1answer
94 views

Bernoulli Trials and Bayes Rule for a Beta Distribution?

So I stumbled upon the following question from Peter Lee's website. Suppose that your prior beliefs about the probability π of success in Bernoulli trials have mean 1/4 and variance 1/48. Give ...
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Conditional distribution of successes in first m independent Bernoulli trials given the total number of successes

This is my attempt: I have, so far, let $Y_{mi} = 1$ if $ith$ trial in first $m$ trials is a success and of course, 0 otherwise. Indeed, $Y_m = \sum_{i=1}^mY_{mi}$. I think this is what I need ...
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Success of Bernoulli trials with different probabilities

If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful? Is there ...
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124 views

Why is a binomial distribution bell-shaped?

I would expect there to be only be values between zero and one (with 0 => failure and 1 => success), but instead the values go up much higher. For example, if I search for "binomial distribution ...
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distribution of the probabliity of multivariate Bernoulli

A p-dimension vector $\mathbf{X}$ has multi-variate Bernoulli distribution and all dimensions are mutually independent. $\mathbf{X}$ has distribution ...
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Combined confidence of hit/miss results from 20 different test subjects

I'm a computer science student and statistics isn't my strong suite. I would appreciate some help. I did a task performance experiment for my Master's thesis to validate my "special secret ...
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A variation of binomial model with gain associated with each success

I am trying to develop my first machine learning model. Each example k in data has [#trials(k), #successes(k), gain(k)]. So total gain for example k is ...
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1answer
47 views

random variable and bernoulli distrbution

I was asked the following question: X is a random variable which follows a Bernoulli distribution with parameter p and take $Y=a+bX$. Compute $\mathbb{E}(Y^3)$. Can anyone please help me.
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Independence in a Joint Bernoulli distribution

Given that [X = 1] and [Y = 1] are independent in a joint Bernoulli distribution, how do we show that X and Y are independent?
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Composite random variables and sufficiency

I wish to find a sufficient statistic for a composite random variable ... Suppose $Z$ is Bernoulli$(p)$ and let $$X|Z=z\sim\begin{cases} N(0,1) & \text{if }z=1\\ E(\lambda)& ...
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Negative binomial distribution & Bernoulli distribution - expected value, MGF

could anyone please help me with the following exercise? Let us assume that the random variable $N$ has a negative binomial distribution $P(N=n) = (n+1)(\frac{3}{4})^2(\frac{1}{4})^n$. $X_1, X_2, ...
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Convergence of Bernoulli sampling procedure

Let $X$ be a variable which has density $f_{X}$ and mean $E[X]=\mu$ Define $B$, a Bernoulli variable, which has a probability $P(b_{i}=1)=p_{i}=\frac{1}{N^\gamma}$ Consider the following ...
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Expected number of successes from $N$ Bernoulli trials with different $p$

Suppose I have N probabilities $(p_1, p_2,...,p_N)$ that represent the chance that each that a corresponding test was passed. How do I apply the Bernoulli distribution to determine the expected number ...
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1answer
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Distribution of partially observable binominal parameter

I suspect this is a textbook question but I don't seem to have the right textbook. Anyway I am trying to estimate probability of coin landing on heads, p, by repeatedly flipping it N times, i.e., ...
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308 views

T-test for Bernoulli Distribution- Sample or Population data for SE calculation?

Am struggling to understand part of the answer to a question have done- Qu- In a given population, 11% of the likely voters are African American. A survey using a simple random sample of 600 landline ...
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Distinguishing two Bernoulli random sources

I have a source of independent Bernoulli random variables, which is either $(X_i)$ or $(Y_i)$ where $\Pr[X_i=1]=p$ and $\Pr[Y_i=1]=q$. I can sample as many values as I'd like, and would like to figure ...
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161 views

Name of single sample multinomial distribution

The Binomial$(n, p)$ distribution is called "Bernoulli distribution" with parameter $p$ in the special case $n=1$. Many properties of the Binomial are derived from the fact that the sum of $n$ i.i.d. ...
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Representative resampling

I am working with a population in which each individual has, among others, 6 observed variables that can be 0 or 1: $X_i \sim Bernoulli(p_i),\ i=1,...,6$ . I know the "true" value for the ...
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Expected value of product of non independent Bernoulli random variables (correlations are known)

I've asked a question about getting the joint probability distribution for $N$ Bernoulli random variables, given the expected value for each one ($E[X_i]=p_i)$ and it's correlations ...
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Hypothesis test based on (random) result of previous experiment: can I multiply confidence levels?

Consider an i.i.d. sequence of Bernoulli random variables $\{Z_i\}$ with Bernoulli parameter $x$. By a certain procedure (experiment A) I obtain a random set S such that $\mathrm{Pr}[x \in S] \geq ...
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Is this correct hierarchical Bernoulli model?

I have a question about correctness of a model that I used for a fairly simple experiment. I'm not sure if it should go to stackoverflow or crossvalidated, because I feel like my question is both ...
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Analyzing binomial distributed variables

I have the following situation. A subject comes to the clinic at day 1 and is evaluated using a 10 item checklist. The sum of those 10 items is the subjects score. A intervention is performed. The ...
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119 views

Joint Probability of correlated Bernoulli distributed Random Variables

I have a Bernoulli source that generate N bits (1/0) with parameter p . I want to find the joint probability of having at most 1 bits = 1 in every m consecutive bits. For example, if the sequence ...
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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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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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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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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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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. ...