The Bernoulli distribution is a discrete distribution parametrized by a single "success" probability. It is a special case of the binomial distribution.

learn more… | top users | synonyms

2
votes
1answer
37 views

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 ...
3
votes
2answers
71 views

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 ...
0
votes
1answer
20 views

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., ...
0
votes
1answer
32 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 ...
1
vote
0answers
20 views

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 ...
4
votes
1answer
152 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. ...
1
vote
1answer
34 views

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 ...
7
votes
1answer
64 views

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 ...
2
votes
0answers
14 views

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 ...
0
votes
0answers
77 views

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 ...
1
vote
1answer
29 views

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 ...
4
votes
1answer
74 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 ...
3
votes
2answers
69 views

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: ...
0
votes
0answers
58 views

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 ...
2
votes
2answers
64 views

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 ...
1
vote
2answers
41 views

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} ...
3
votes
1answer
49 views

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 ...
3
votes
1answer
41 views

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 ...
2
votes
1answer
18 views

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 ...
1
vote
0answers
50 views

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. ...
3
votes
1answer
48 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 ...
3
votes
2answers
134 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 $X$?
7
votes
1answer
486 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. ...
3
votes
1answer
126 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 ...
2
votes
2answers
135 views

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. ...
2
votes
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 ...
0
votes
0answers
17 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. ...
2
votes
0answers
36 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 ...
1
vote
0answers
13 views

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 ...
4
votes
1answer
46 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 ...
1
vote
1answer
40 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 ...
0
votes
2answers
750 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 ...
0
votes
0answers
17 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 ...
6
votes
2answers
147 views

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 ...
2
votes
0answers
197 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 ...
8
votes
2answers
217 views

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} ...
2
votes
1answer
112 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 ...
1
vote
1answer
184 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 ...
1
vote
1answer
48 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 ...
2
votes
1answer
80 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)}] + ...
3
votes
1answer
107 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 ...
10
votes
4answers
2k views

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 ...
0
votes
1answer
31 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 ...
3
votes
0answers
23 views

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. ...
1
vote
1answer
49 views

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 ...
2
votes
1answer
210 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 ...
2
votes
1answer
137 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 ...
1
vote
1answer
27 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 ...
5
votes
4answers
6k views

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 ...
1
vote
1answer
120 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 ...