# Tagged Questions

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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### Use full or reduced model to identify significant variables to then determine pairwise differences?

I have a hypothesis testing model in which I would like to know if environmental variables I collected influences the probability a bacteria on a given amphibian kills a pathogen. Following ...
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### R - (why) does fitting a binomial glm to a 3-level factor work?

I am trying to determine which of several independent variables {A,B,C} best predicts a subject's response R, which can be one of 3 choices, {high, medium, low}. As the dependent variable is ...
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### Distribution of combination of events with different distributions themselves

I am asked to find the PMF of $X$, for the following definition of $X$: "There are two coins, one with probability $p_1$ of Heads and the other with probability $p_2$ of Heads. One of the coins is ...
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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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### how to fit a generalized nonlinear model in R?

I've been trying to find a way to fit a generalized nonlinear model in R, with little success. The stochastic component would be the following: $$Y \sim binomial(n, pi)$$ The parameters being b1 and ...
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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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### Exponential Distribution with possible Binomial Probability

You have a system with 6 components. In order for the system to work the following must be met: Component 1 must work. At least one of components 2, 3, 4 must work. At least one of components 5, 6 ...
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### Validation of linear logistic regression for binned binomial data

I am looking at how pavement thickness variation parameters(absolute thickness, variance, slopes, critical points) relate to distress occurrence. After some research, I have settled on the linear ...
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### Disagreement between the p-value and the confidence interval

This is a question regarding the binomial test in R. What happens if I get a p-value < 0.05, which would generally cause me to reject the null hypothesis that the population value is equal to a ...
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### Sum of Independent Binomials

Let $X \sim \text{Bin}(m,p)$ and $Y \sim \text{Bin}(n,p)$. I want to find the distribution of $W = X + Y$ using a convolution theorem that says: $$f_W(w) = \sum\limits_x f_X (x) f_Y(w-x).$$ So using ...
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### Estimate population parameters base on two-level random sampling

I have a general population(N) and then random-sampled n samples from N. For each individual object in the samples, I random-sampled 1000 activities(for example, let's say 1000 "online display ads") ...
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### How to derive Clopper-Pearson interval's F and beta approximation?

It is well-known that there is an approximation of the Clopper-Pearson exact Confident Interval for binomial test. On Wikipedia it just simply claimed, without any reference that: Because of a ...
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### Convert probabilities of “binomial events” into beta distribution for binomial parameter (theta)

I have three events that essentially arise from a binomial distribution with k=2, i.e., X~Binomial(theta,n=2). Given some model and data I obtained a conditional (pos distribution for a particular ...
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### Collaborative filtering when the ratings are estimates

I have a matrix (User X Item) of binomial estimates of click through rates. That is, each entry in the matrix is the estimate of the probability that a specific user will click on a specific target, ...
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### Binomial, as N goes to infinity

I live in a world of size $N$. There are $U = uN$ many urns, and $B = bN$ many balls. The likelihood of any particular urn having $k$ many balls is given by $f_N(k, n=bN, p=(uN)^{-1})$, where $f_n()$ ...
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### Test of strict difference in independent binomial probabilities

Suppose $X,Y$ are two independent binomial random variables with parameters $n_1,p_1$ and $n_2,p_2$ respectively. Suppose one wanted to test the hypothesis $p_1>p_2$. Conditional on $X+Y=s$, ...
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### Does sampling from a larger sample change how much variance around an estimate to expect?

Suppose you have one billion marbles. Each marble is either red or white. Call the proportion that are red, p. You randomly sample 100 marbles. Based on this sample you estimate $\hat{p}$. You can ...
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### Statistical validity of association rules with low support

We have data sample with 10,000 observations. We are using association analysis to find segments with low support (< 4%) and high(er) confidence. Our rules are always in such way that on the right ...
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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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### How do you deal with overdispersion in a zero-inflated negative binomial regression AND when you expect data to have zeros?

Background: I am analyzing the effect of multiple variables (lineage, ancestral plant species, plant species reared from, larval density, body mass) on different traits: ovigeny index (initial egg ...
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### Test of Equal Proportions with exact p-values [duplicate]

The question "Test of Equal Proportions with zero successes" showed me that using a test of equal proportions with very low or very high numbers of successes doesn't give an accurate p-value. Below is ...
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### Fitting a Fractional Logistic Model using GLM in R

I have a response variable that is a proportion bound between 0 and 1. I would like to fit a fractional logit model using R. I've consulted a few posts on crossvalidated and stackexchange and came up ...
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### Test for difference in proportions with small probabilities

I want to perform a test for differences between 2 binomial populations. The probabilities are small (usually less than 10 %). I can define the successes as "rare events". I think that the z-test ...
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### Convergence of moments of binomial to Poisson

As is well known, the $\mathsf{Binomial}(n,p)$ distribution converges to the $\mathsf{Poisson}(a)$ distribution as $n\rightarrow \infty$, $p\rightarrow 0$ with $np=a$. I'm pretty sure that the ...
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### Why are there huge differences in the SEs from binomial & linear regression?

I have data from a simple experiments where people put (a fixed number of) balls either to the left or to the right of them (each ball is just the same with regards to consequences of putting them to ...
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### Comparing likelihoods of different experiments

Call $X_{i,j} \sim B(p_{i,j})$ a set of independent coin tosses with parameters $\{p_{i,j}\}_{i,j}$ where $j\in[1,p]$ and $i\in[1,n]$. In plain text, all the coins are biased in a different way. ...
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### Can I do a Binomial test on more than one individual/group? Can a repeated t test have different sample sizes?

I've conducted an experiment on chickens, training them to associate either blue or yellow with a 30 or 1 second wait before a food reward and then giving them a preference test, 15 times in total and ...
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### Probability of observing an excess of $k_0$ points via a binomial distribution with the isotropy assumption

Consider an area $\Omega$ and its subset $\Omega_1$. Let the hypothesis be that the distribution of points in the area $\Omega$ is isotropic. That means that a probability of finding a point in the ...
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### what's the right way to combine binomial parameter estimates?

let's say you have a bunch of weighted coins. by observing flips, you can estimate (a confidence interval for) the binomial parameter for each. how should you combine these to summarize the ...
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### Is the Standard Deviation of a binomial dataset informative?

I am working on a dataset of presence/absence data, with my response variable being 'proportion of sites where X is present'. I have been asked to provide standard deviations alongside the mean ...
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### Bayesian inference with unequal sampling

I have a "two-column" data set, with a multi-class categorical variable A, and two-class variable B. It is assumed that each observation is independent. For each category of variable $A$, I want to ...
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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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### PCA on count-based data

I'm looking to do a PCA analysis on count based data itself rather than averages. I'm hoping this will help for variable observation depths; for example, 3/4 reads is not really equivalent to 15/20. ...
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### 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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### Can I use Bayes Theorem to find a conditional distribution rather than conditional probability?

I might be going about this the wrong way, but I'm trying to develop an understanding of a particular conditional value, say $P(CustomerBuysFries | CustomerBuysHamburger) = P(F|H)$. Ultimately, I want ...
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### Variance of the logarithm of a binomial distributed random variable

What is the variance of $\log(1+X)$, where $X \sim Bin(n,p)$. I am looking for an approximate solutions depending on n and p. I have tried Taylor expanding the logarithm, but it didn't lead to a ...
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