Questions tagged [bernoulli-distribution]

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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Test for a Bernoulli model with a single trial

I have a model that predicts Bernoulli probability parameters $p_i$ at $i\in[1..100]$ sites. To test this model, i can only take one trial at each one of the sites, resulting in $\approx10$ successes....
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Central limit theorem for dependent binary-related variable

Let $Y\sim N(\mu, \sigma^2)$ and given sample size $n$, we have an iid sample $\{Y_1, ..., Y_n\}$. We sample $X$ (size $n$) from Bernoulli with probability $\pi$. Denote $Z_i=X_iY_i$. Then, when $X_i=...
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$Y$ has uniform distribution on [0,1], and conditional on Y = y, let X have a distribution of Bernoulli(y). What's P(Y|X=1)?

Using Baye's formula I have $P(Y=y|X=1) = \frac{P(X=1|Y=y)*P(Y=y)}{P(X=1)}=\frac{y*1}{P(X=1)}$ Now $P(X=1) = P(X=1|Y=0)P(=0)+P(X=1|Y=0.001)P(Y=0.001)+...P(X=1|Y=1)P(Y=1) = 0+0.001+0.002...+1 = \int_0^...
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How can I derive the EM algorithm for a mixture of two Bernoulli distributions?

How can I derive the E-step and M-step in the EM algorithm for a mixture of two Bernoulli distributions? Note that I am aware that there are several notes online that explain how to do this for the ...
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Why does redrawing a sample of a binary variable split into two groups generate different distributions according to prop.test()?

I was working on some power calculations and stumbled upon results I don't understand. Say we have a sample of a binary variable a, and the sample probability of success is p. We randomly split the ...
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How to find the median of Bernoulli trials and $\bar{X} = (X_1 + X_2 + X_3)/3$?

Assume $X_{1}, X_{2}$ and $X_{3}$ are independent Bernoulli trials with $p=0.5 .$ Let $\bar{X}=\left(X_{1}+X_{2}+X_{3}\right) / 3$ and $M=\operatorname{median}\left(X_{1}, X_{2}, X_{3}\right)$. (a) ...
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Absolute of expected value of multivariate correlated Bernoulli

I am running some experiment where I draw samples from a multivariate Bernoulli distribution (in this case taking values -1 or +1) with a single correlation coefficient (i.e., same correlation for all ...
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Efficiency of two estimators for a sample from a Bernoulli population

Given a Bernoulli population, I have two estimators for a random sample of size $n$: $T_1=\frac{\sum\limits_{i=1}^n X_i + 2X_n}{n+2}$ $T_2=\frac{\sum\limits_{i=1}^{n-2} X_i + 2X_n}{n+2}$ I want to ...
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What would be the proper distribution to model the number of particles in a state in canonical ensemble

Suppose my system has $N$ particles, and I want to find a distribution for $n_i$, the number of particles in the $\epsilon_i$ energy state. What I do know is the boltzmann probability, which tells me ...
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Bayesian inference - Calculating the prior distribution of the parameter in the Bernouli distribution from a series of bernouli proccesses

What I have are n different time series of bernouli processes of varying lengths, taking the values of 0 or 1. What I would like to do is to use Bayesian inference to calculate, for one of these ...
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30 views

Confindence Intervals for Ordinal Variable

I count how many days within a month people have executed a certain activity. The maximum is 30 (i.e., the person executed the activity each day), the minimum is 0 (i.e., the person never executed the ...
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Hypothesis Testing on Derived Distributions

Suppose we have access to samples from two probability distributions $P$ and $Q$ which may be arbitrary and high dimensional but are over the same domain $\mathbb{X}$ (for example $P$ and $Q$ may be ...
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Difference between multinomial distribution with n trials and categorical distribution performed n times

I want to understand if there is any difference between performing multinomial distribution with 1 trial, 10000 times and performing multinomial distribution with 10000 trials, 1 time. Here is the ...
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Using Monte Carlo to sample from marginal distribution

I am defining a model on a vector, $T$, of size $n$, wherein each element $t_i \in T$ is independent and either $0$ or $1$. This model depends on 3 other parameters, $q$ (also a vector of size $n$), $\...
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Application of spike and slab for sampling from posterior distribution (bernoulli and beta)

I think the gamma N term in the first equation relates to the spike and prior. However, I am unsure what the rhs of the first is used for? Further, I am unsure what the pie term of the second equation ...
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Compute log-likelihood in Bernoulli Gaussian Mixture

I'm working on this exercise about Gaussian Mixtures: Here's part of the solution: I don't understand how they came up with this equation for the log-likelihood (red arrow). From what I know, the ...
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Is there any distribution that only takes value 0 and 10, both have 1/2 probability?

Now I want to generate random numbers. It has 1/2 probability generating 0 and 1/2 probability generating 10. So what the distribution of the numbers is? It looks like Bernoulli but it isn't. (...
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Transforming the expected value of $Y_i$ in binomial regression

Currently, I'm learning generalized linear regression (GLM). There is something troubling me concerning binomial regression. In this text, in the part about the structure of a GLM, the random ...
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Implicit Ordering in Bernoulli Distributions?

Let $X$ be a Bernoulli random variable with success parameter $p$ As we usually consider the support as $\{0, 1\}$, we write $P(X = 1) = p$ and $P(X = 0) = 1-p$ This $0,1$-encoding seems to induce an ...
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Normal approximation to Bernoulli variable

I'm looking for a normal approximation for a Bernoulli variable (so I can later sum multiple correlated approximated variables) The trivial approximation is taking the mean and variance of the ...
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Exponential transformation of Bernuolli variable

Assuming that $x$ is a Bernoulli random variable with probability $p$, let $Y = 2^X3^{(1-X)}$, What is the pmf of $Y$ What is $E[Y]$ What is $Var[Y]$ I have never seen an exponential transformation ...
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Conditional Density Of Independent Bernoulli Random Variables Given Their Sum

Let Yi's be m independent Bernoulli random variables with corresponding success probabilities pi's, and let S = sum of Yi's. I am trying to figure out a way to find the given conditional probability, ...
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How to estimate multivariate Bernoulli with missing labels

I want to estimate a multivariate Bernoulli distribution with missing labels, without assuming independence. The dataset is from https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/shape/...
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Trouble reading bernoulli Naive bayes notation [duplicate]

Here is a mathematical description of the bernoulli naive bayes taken from the book ,Bayesian Reasoning and Machine Learning by David Barber i want to know what does the notation below means ? if ...
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What is the variance of a constant and set rate?

Say we have two options: out of every 25 videos, one will be promoted every video has a 4% chance of being promoted For the first example, it's meant that after 25 videos, it's guaranteed that ...
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Joint distribution of data sampled iid from a bernoulli process, and the absence of binomial coefficient

Let's assume I have 4 observations with each observation is modelled as a bernoulli trial with probability $p$. Sucesses are labelled as 1, failure is 0. My observations $(x_1, x_2, x_3, x_4)$ are as ...
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Properties of the sample mode for Bernoulli data

Suppose we have a sample $X_1,...,X_n \sim \text{IID Bern}(p)$ of Bernoulli values with probability parameter $p \neq 0.5$. Denoting the sample proportion $\hat{p}_n$ we define the sample mode as: $$\...
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Question regarding binomial and bernoulli notation

I have a bunch of independent binary variables Xk where k >= 1 and P[Xk = x] = 1/4^u * 3/4^(1-u) for u = 0 or u = 1 ... this is a bernoulli distribution, right? I'm confused on whether I should ...
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104 views

Newton's method for Bernouilli likelihood with ridge penalty

I am trying to derive the gradient and hessian of logistic regression with ridge penalty. The log-likelihood should be (correct me if I am wrong): $$\sum_{i=0}^n\Big(\log{(P_i^{y_i}(1-P_i)^{1-y_i}- \...
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Bayesian (continuous) logistic regression model with Beta likelihood?

I have a problem where my target variable are continuous/float values in the range [0,1]. If my data were integers in {0,1} this would be a simple logistic regression / Bernoulli likelihood problem. ...
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Estimating Joint Probability of non-i.i.d., dependent, Bernouli trials

A joint probability space of $n$ bernouli trials has $2^n$ parameters (one probability for each configuration of T/F). I want to estimate the distribution using a subset of the necessary knowns. ...
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Sample three Bernoulli variables given the 2 by 2 by 2 table

I want to sample three Bernoulli variables given their 2 by 2 by 2 table: where the last column is the probabilities with sum being equal to 1. What I am thinking now is to use a multivariate normal ...
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Simplify Equation with Random Variables

I'm wondering if whether the following problem has a solution. Suppose we have i random variables, all independent, and all following a Bernoulli distribution with parameter $p_i$ (all $p_i$'s are ...
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67 views

how to sample from a conditional Bernoulli distribution

Given a variable $x_t \in \{0,1\}$, then we sample $x_{t+1}$ in the following way $$ x_{t+1} = x_{tmp} x_t + (1-x_{tmp})(1-x_t ), \ x_{tmp} \sim Ber(x_{tmp};p). $$ Does $x_{t+1}$ follows the ...
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Bernoulli distribution

Definition 3.3.1 (Bernoulli distribution). An r.v. X is said to have the Bernoulli distribution with parameter p if P(X = 1) = p and P(X = 0) = 1 − p, where 0 < p < 1. We write this as X ~ Bern(...
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Suppose $Y\sim N(0,X^2)$, and $X\sim Bernoulli(0.5)$ on $\{1,2\}$, how to prove (or rigorously argue) that $X,Y$ are not independent

Suppose $Y\sim N(0,X^2)$, and $X\sim Bernoulli(0.5)$ on $\{1,2\}$, how to prove (or rigorously argue) that $X$ and $Y$ are not independent? Can I proceed as follows: try to show that there exists $x$ ...
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Expectation and Variance of Sum of dependent discrete variables

Q. Let X be a discrete random variable such that X = 0 with probability 0.5 and X = 1 with probability 0.5. Let Y be a discrete random variable such that Y = 1 when X = 1 and Y = 0 when X = 0. What is ...
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Logic of Sklearn Bernoulli Naive Bayes Classifier when the the predictors are not even binary?

I know the mathematics behind the Naive Baye's Bernoulli Classifier Algorithm and it is used to calculate the probabilistic results. As we know the Bernoulli Naive Bayes Classifier uses binary ...
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62 views

Distribution of sum of possibly non-independent Bernoulli random variables with known variance-covariance matrix

I wonder if there are any results concerning the distribution of sums of possibly non-IID Bernoulli random variables when covariances in all pairs of r.v.'s are known. To make this more concrete ...
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Draw two balls without replacement

Suppose we consider an urn with 3 red balls and 5 blue balls. We now draw two balls without replacement. If we draw a red, it is a success otherwise a failure. Let X=1 if we draw a red ball in the ...
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Dependent Bernoulli trials with same probability of success? [duplicate]

I am not that good at probability theory. My need: I need a simple example of a series of Bernoulli trials with the same probability of success in each trial, but where the Bernoulli trials are ...
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Parrondo's Paradox game using frequency instead of probability

Given the following probabilities : Game A : coin flip (p1 = 0.49) Game B1 : coin flip (p2 = 0.09) Game B2 : coin flip (p3 = 0.74) Given the following game : ...
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Checking if a coin is fair: z-test or t-test

I have seen many questions on this topic, but none of them could answer my question. Suppose I flip a coin 1,000 times and got 490 heads. I want to test if the coin is fair. I don't want to use the ...
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Probability question - Bernoulli trials vs Binomial distribution

I am handed a fair coin. I understand that the probability of H/T is 0.5. Trial 1. I flip and observe H. I expect that on my 2nd toss, the probability of H is still 0.5 Trial 2. I flip and observe H. ...
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Calculating $p$ for mixture of Bernoulli distributions

I was reading this from University of Buffalo Mixture of Bernoulli lecture slides. So is the new $p_k$ or as they denote it $\mu_k$ for Bernoulli dist $k$ just the mean of all the points multiplied ...
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Probabilty estimation for Bernoulli with number of trials as random variable

Problem description Suppose we have fixed number of people that are the test population, let's say $t=200$ persons. For each one of them $\mathbf{r}_j$ we know about $m=300$ features that describes ...
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Bernoulli distribution; test p <= 0.3

I have a random variable $X \sim Ber(p)$ and I should test: $$H_0: p \leq 0.3$$ $$H_1: p > 0.3 - alternative$$ I tried to use $\chi^2$- test, but there is a problem: the number of degrees of ...
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78 views

Calculate maximum likelihood function for $\theta$ given data

Say you have a coin A that has probability of $\theta$ of landing on heads and a coin B with probability of $2\theta$ of landing heads. Then say we flipped A 7 times and the first 5 flips were tails ...
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86 views

How to prove that a mixture of Gaussians is Gaussian? [closed]

Consider independent random variables $\mathbb{P}(X=-1) = \mathbb{P}(X=1) = 0.5$ and $Y \sim \text{N}(0,1)$ and let $c \neq 0$ be a real constant. How do I prove that $cXY$ is Gaussian ? What is the ...
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Test whether probabilities of heads are under-estimated across many coins

I am trying to figure out a problem that is equivalent to the following. Suppose you have a bag of $n$ coins in which each coin is labelled with a probability $p_i$ that it will come up heads when you ...

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