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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Calculating prediction variance from JAGS model of a bernoulli outcome in R

I have a model of a bernoulli random process I fit using JAGS via the rjags package in R. Here are some example data, as well as code to fit the given models in ...
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44 views

Calculating conditional probability in Bernoulli mixture model

I'm following the book Pattern recognition and machine learning by Bishop on Bernoulli mixture model, and trying to code it. But I don't understand how to calculate the conditional probability (page ...
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22 views

If X is Beta distributed, what is distribution of Y: Y=1 for X>0.5 and Y=0 otherwise

Consider random variable $X \sim \mathrm{Beta}(\alpha,\beta)$ What is the distribution of $Y$ defined by $Y\in \{0,1\}$, $Y=1$ if $X>0.5$ and $Y=0$ otherwise.
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Problem Calculating Standard Error

I have a set of individuals categorized into two classes as follows: ...
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Binary outcome estimation from n “measurements”

Assume you have a binary parameter $y$ which is either $0$ or $1$, and have $n$ measurements $\tilde{y_k}$ of it, each with a different probability $p_k$ of being correct (equal to $y$). With $k=1,..,...
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Skewness 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. Let's define the random variable $S$ which is the ...
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19 views

Bernoulli hypothesis test where null hypothesis is that $p$ does not equal $p_0$

I am trying to find a test, that tests if a random variable $X$ is Bernoulli distributed with $p=p_0$. However I want the hypotheses to be $H_0$:$ p \neq p_0 $ against $H_1: p = p_0$.
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1answer
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analyzing binary mortality data collected at different remeasurement intervals. Bonus for R implementation

I am replicating an analysis that models tree mortality data. Data are structured such that forest sites are revisted at some random interval, which is recorded. It is then determined if a tree lived ...
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18 views

How many boxes have balls?

I have N boxes, each of which may or may not contain a ball. I want to know how many boxes have balls by sampling the boxes in the following way. For each box in the N boxes: I open this box with ...
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409 views

Simulate a Bernoulli variable with probability ${a\over b}$ using a biased coin

Can someone tell me how to simulate $\mathrm{Bernoulli}\left({a\over b}\right)$, where $a,b\in \mathbb{N}$, using a coin toss (as many times as you require) with $P(H)=p$ ? I was thinking of using ...
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How to derive bernoulli deviance

I found the deviance definition in https://en.wikipedia.org/wiki/Deviance_(statistics) and the one-observation bernoulli deviance in Scikit Binomial Deviance Loss Function. $$ \text{bernoulliDeviance}...
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Standard Error for Proportion of Successes in Monte Carlo Simulation

First note, this is for an assignment. I've been through all our notes, researched online and still unsure on this. We are asked to run a stochastic simulation where, at the end of each run, there is ...
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45 views

observed data log likelihood and complete data log likelihood estimation

I have the following model. $p(z_{n}) = Categorical(\pi)$ $p(\pi) = Dir(\alpha)$ $p(x_{n}| z_{n}=k,\mu) = \prod_{d=1}^{D} {Bernouli(\mu_{kd})} $ $p(\mu_{kd}) = uniform(0,1)$ How can I find ...
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310 views

Why 0 for failure and 1 for success in a Bernoulli distribution?

Why do we choose these numbers and not for e.g $e$ and $\pi$ for success and failure in an experiment? What is the logic and how badly would it affect my calculations if I choose other values instead ...
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1answer
31 views

Standard deviation of a Bernoulli distribution?

Example: We have a population of 100 people, where only 60 of them love pizza. So, the probability of success is 0.6 SD[x] = $\sqrt{pq} = \sqrt{0.6 \times 0.4} = 0.48 $ The definition of SD is : ...
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19 views

Hamming distance of Bernoulli RV

Assume you draw $k$ data points out of $n$ data points. Each data point is composed of $m$ Bernoulli random variables. You may assume that the data points are i.i.d and likewise their coordinates, but ...
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23 views

Probability of n consecutive sucesses in N independent Bernoulli trials

How can one compute the probability of n consecutive successes in N Bernoulli trials ? (n < N)
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Calculating confidence interval for whether some of N coins are unfair

I'm having trouble figuring out the appropriate way to place a confidence interval on a ratio I'm estimating. It's for a social science application, but I'll present the problem via coin-flips as I ...
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Estimating the probability in a Bernoulli process by sampling until 10 failures: is it biased?

Suppose we have a Bernoulli process with failure probability $q$ (which will be small, say, $q \leq 0.01$) from which we sample until we encounter $10$ failures. We thereby estimate the probability ...
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1answer
18 views

expected value of filtered binomial

Assume a bernoulli random variable with success probability p. In 12 independent trials of the bernoulli I only look at the third, sixth, nineth and twelfth experiment. What is the expected value of ...
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1answer
30 views

Probability of a success at the k-th trial in N repeated Bernoulli trials

For a Bernoulli random variable with success probability $p$, how does one compute the probability of having a ''success'' at the $k$-th trial out of $N$ repeated and independent trials where $k \le N$...
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1answer
37 views

estimating a parametric function

I am working on this problem and am stumped. Can anyone take a look at it? $X_1, \ldots ,X_n$ are distributed Bernoulli$(p)$ where $p$ is unknown. Consider the parametric function $\tau(p)=(p+(1-p)...
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Combining Exponential and Bernoulli distributions

I am building a model of customer spending, and need some help to identify the best way of doing this. (1) I have an exponential distribution for each customer's spending (2) Bernoulli distribution ...
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32 views

Expectation from conditional expectations

Consider a real-valued random variable $X$ and a dichotomus random variable $Y\sim Be(p)$, all defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Could you help me to show ...
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Probability of non recurrence given a particular time of non-reccuring

I've been having a cardiac problem that may (or may not) be linked to alcohol consumption. If its happening about once a week on "social use", how long do I have to be completely alcohol free and ...
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113 views

How to calculate one probability out of several Bernoulli probabilities?

For a multi-agent problem I want to calculate the probability that a certain event happens for any agent (1 or more). There are n agents, and $P(X_i=1)=p_i$ for each agent. I want to calculate the ...
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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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1answer
44 views

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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33 views

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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1answer
31 views

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 $\mathbb{E}(X)$...
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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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37 views

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: $$\mathbb{E}[X+1]\mathbb{E}\...
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43 views

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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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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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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1answer
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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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1answer
223 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 $p(X)=\Pi_{i=1}^pp_i^{X_i}(1-p_i)^{1-X_i},\;X_i\in\...
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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 algorithm"....
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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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63 views

Expectation of a cubic transformation of a Bernoulli random variable

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 $\mathbb{E}(Y^3)$.
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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)& \...