# Questions tagged [binomial]

The binomial distribution gives the frequencies of "successes" in a fixed number of independent "trials". Use this tag for questions about data that might be binomially distributed or for questions about the theory of this distribution.

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### Determining the sampling size ensure defect rate of less than X parts per million at a confidence interval

For quality sampling, I need to ensure a defect rate of less than 5000 parts per million, which is 1/200, at a specific confidence level. The goal is to have the smallest sample number for the ...
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### Maximum likelihood estimator of $n$ when $X \sim \mathrm{Bin}(n,p)$

Given a random variable $X\sim Bin(n,p)$, where $p$ is known $p\in (0,1)$ , $n$ is an unknown positive integer and $x\in\{0,1,2,....n\}$, what is the maximum likelihood estimator of $n$? I ...
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### Do I add both the new variable and the original variable to my log binomial model?

I'm calculating interaction indices for some variables in my model. I have an interaction between two variables (age (continuous) and hypertension (binary)), and I read in a similar publication that ...
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### Binomial proportion confidence interval when rigid bounds on population proportion (0.5 < p < 1)

A standard problem is estimation of confidence intervals for a population proportion given that one has observed f successes out of n independent trials. There is a reasonable discussion of this ...
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### How do I interpret this interaction between a continuous variable and a binary variable from a log binomial model?

I'm new to modelling and not sure how to interpret this interaction for a results section. The model output tells me that the interaction between cancer (binary) and age (continuous - one year ...
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### Analyzing repeatability using rptR package with binary data, CI doesn't match estimated R [closed]

I'm using the rptR package in R with the following code: rpt(SUCCESS ~ TRIAL + (1|SUBJECT), data=phase1, datatype ="Binary", grname="SUBJECT", nboot = 100, npermut = 0, ratio = TRUE) I've also ran ...
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### How do I determine goodness of fit for a test of n binary decisions?

Let's say I have $n$ pairs of items. For each pair I select one item over another due to some perceived trait. This could be anything, for example, identifying which image is brighter than the other ...
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### What is the probabilty that X > 2 conditioning on Y = 2? (Homework)

another homework question here. Let 𝑌 be a binomial random variable with 10 number of trials and 0.2 probability of success. Let X be a uniformly distributed random variable over the interval [0, 3]. ...
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### (Binomal Distribution) With n = 20, p = 0.2, calculate the probability that 2X >4 (Homework) [duplicate]

This is a homework question. The scenario is as follows: A dishwasher can accommodate 20 plates in one loading. With every loading, there is a 20% chance that 1 plate will not be cleaned properly. ...
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### Are the beta distribution and binomial distribution related? [closed]

I've seen questions like this and this, but it hasn't quite answered my question. How intertwined are the Beta and Binomial distributions? A quick sidenote: the Poisson distribution and the ...
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### Degrees of freedom - binary vector

I came across the following statement: A general distribution over a binary vector (of lenght $N$) has $(2^{N}-1)$ degrees of freedom, whereas a factorial distribution over a binary vector has ...
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### Statistical confidence in voting samples

I'm currently watching the USA elections and I'm wondering how the confidence is calculated. So let us say we have a state of 500,000 inhabitants; 5,000 votes have been counted and 80% (4,000 votes) ...
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### Chi-squared test and binomial distribution

In some circumstances the Chi-square test can be replaced by a test based on the binomial distribution. The common issue with using the binomial distribution is that it may quickly become tricky to ...
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### Does a 2*2 chi-square test give the same results as a binomial test?

This is a conceptual question. Suppose we have an experimental design with two grouping variables (e.g., the subjects are either male or female) and the variable we are measuring is a binary variable, ...
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### How to find P(2X>4)?

This is a Binomial question. I am required to find P(2X>4), and given that X~B(10,0.2), how do I convert it to 2X~B(n,p)? Have tried converting from P(2X>4) to P(X>2), but understand that it doesn't ...
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### Am I understanding how a Confidence Interval is obtained?

In short: I'm wondering if I'm understanding the concept of two-sided "Confidence Interval" via an example? Details: Suppose we observe 5 successes in 20 trials and so our observed $p = 5/20$. To ...
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### How to combine/pool binomial confidence intervals after multiple imputation?

After I multiply imputed my dataset m times I wanted to calculate a binomial proportion confidence interval. How I can I combine the various estimates of the confidence intervals while taking Rubins ...
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### Binomial distribution where probability of success is dependent on another binomial distribution

How does one model the Binomial distribution where the probability of success is the result of another Binomial distribution. For example, say I make 10 coin tosses many times and record the number ...
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### Equivalence test for one proportions test

I want to apply an equivalence test on my sample to infer whether they are equivalent or not. Since my data are binominal [0,1] I don’t know whether the TOST procedure (tost() in R) can handle my ...
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### Limiting distribution of order statistics with binomial weights

Let $G$ be a CDF whose support is $[0,1]$ and $x\in(0,1)$ is a constant. Define a CDF $H^n$ by $$H^n(t)=\sum^n_{k=1}{n-1\choose k-1}x^{k-1}(1-x)^{n-k}G_{k;n}(t).$$ where $G_{k;n}$ is a CDF of k-th ...
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### Test if two binomial distribution are significantly different from each other

I obtained two groups of data from my experiment and calculated the binomial distribution for each group, trying to see if they are significantly different from each other. However, the z sore I got ...
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### Bernoulli Cusum Chart?

Can I use the R package named 'cusum' to make bernoulli cusum charts or should I use another package? I've been asked to explore using Bernoulli cusum chart for trending a parameter that's usually 0 ...
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### Probability statistics - probability that 10 or more of the donors are Rh-negative?

In a certain population, 18% of the people have Rh-negative blood. A blood bank serving this population receives 95 blood donors on a particular day. So my teacher posted a solution to a problem ...
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### Can multinomial distribution be simulated by a sequence of binomial draws?

I wonder if single multinomial distribution (I will use the notation from JAGS/WinBUGS, but in fact, this is principial thing rather than of particular language) ...
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### When are real limits used for calculating z score?

I'm taking an introductory statistics course. At first, the textbook talks about real limits in the context of continuous variables and frequency distribution table, that is all clear. But on what ...
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### Is Rule of Three inappropriate in some cases?

I'm building binomial proportion confidence intervals for a patient dataset containing the frequency of home nursing visits during the week prior to hospital admission. The freq. of home visit ...
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### binomial pairs probability

A group of four people is said to be “interesting", if there are at most five pairs who are friends. Assume that each pair of people are friends, independent of every other pair, with probability 1/2 ....
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### difference between independent binomial variables

let and $X_1\ and \ \ X_2$ be independent such that $X_1$ ~ binomial (m,1/2) and $X_2$ ~ binomial (n,1/2) , n $\ne$ m then how to prove that $X_1 - X_2 + m$ ~ binomial (m + n ,1\2) ...
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### binomial correlation?

I am interested in testing for a correlation between two variables, both of which are binomial. I guess this is equivalent to a Model II regression where both variables are binomial. Ideally I want ...
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### How to “fit” the size of a binomial distribution

I'm solving a problem where I have 15 samples of a unknown distribution. They ask me to fit somehow the parameters of the Binomial, Poisson and Normal distributions. I could use the sample mean and ...
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### Tuning the “strength” of updates to a posterior distribution for conjugates

I'm asking this question as a sanity check- I am not a statistician or research scientist, and just am doing a gut check on a model I am building. I want to quantify uncertainty of a specific metric ...
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### Calculate the number of trials needed to achieve certain number of successes at a confidence level

If I know the probability of success for an event, the number of time that I want to observe the event at a certain confidence level how do I calculate the number of trials needed?
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### Why “sum of squared Pearson residuals” is around “number of dependent variables” in Binomial distribution?

A Pearson residual is defined as: $r_{i}(\theta)=\frac{y_{i}-E(y_{i}|\theta)}{\sqrt{Var(y_{i}|\theta)}}\tag{1}$ Sum of squared standard residuals $X^{2}$ is: $X^{2}=\sum_ir_i^2\tag{2}$ where $y_{i}$ ...
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### Residual Deviance and degrees of freedom - Negative Binomial Distribution

I am trying to model count data using python's statsmodels module (Beer's sold at a football stadium as function of visitors, "tilskuer", and weather data). ...
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### How to tell the probability of failure if there were no failures?

I was wondering if there is a way to tell the probability of something failing (a product) if we have 100,000 products in the field for 1 year and with no failures? What is the probability that one of ...
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### Testing for extinction using the binomial distribution

I'm using quadrat sampling to determine whether a species of plant is present in a given area - my end goal is to work out how many quadrats need to be checked without finding anything before I can ...
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### Covariance between two binomial random variables

Consider a binomial random variable $X$ with parameter $p$ and another binomial random variable $Y$ with parameter $q$. What is the covariance of $X$ and $Y$? How well does the proof generalize to $n$...
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### Null hypothesis test for proportions without minimum requirements? [duplicate]

I have a typical dataset: sample A (method A) which has 200 samples and 25 failures. sample B (method B) which has 240 samples and 0-10 failures. Note that 0 failures is possible. Null hypothesis is ...
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### How do you calculate an exact two-tailed P-value using binomial distribution? [closed]

First, I will preface this question with my ulterior motive: I would like more evidence that the use of 19th and 20th century approximations play little to no pedagogic advantage in modern intro stats ...
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### flipping card with increasing probability

In an online game you can buy cards which has a 5% probability to be rare cards. If you don't have the rare card for more than 20 times, the chance of next purchase will increase by 5%, repeat until ...
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### truncated binomial samples with GLM

We have a binomial process that yields samples of 60 trials. To save time, once 2 failures have been observed the process is reset. So if a test series hits 2 failures early, the resultant sample ...
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### How can Bayes avoid Cromwell?

I'm studying widgets and their failures. Generally a widget will run for many years without trouble, but 1-2% of widgets will fail in a given year. I have a table which lists widget manufacturers (A, ...
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### Does the number of trials and probability in binomial distribution should be independent variables?

Let's say that the number of trials and probability of success are a function of some variable t i.e, N(t) = f(t) and p(t)=g(t). Can we still use a binomial distribution in this case?
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### Binomial Distribution variable classification

Situation: A coin has a probability for heads p=0.5 on any independent flip. After flipping the coin N=2,500 times, heads occurred G=53% of the flips. Question: The change in which parameter (only 1 ...
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### Relationship between Binomial and Beta distributions

I'm more of a programmer than a statistician, so I hope this question isn't too naive. It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's ...
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### Comparing proportions of two samples in time

I am trying to compare if the proportion of one bird subspecies in a roosting area is different now from over 20 years ago. My dataset looks like following: -In 1996 birds were counted at a ...