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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Best proportions test based on counts (accuracy, distribution, etc)

I'm trying to compare two sets of counts over totals. For example, subject 1A will have 40 out of 600, and subject 1B will have 3 out of 20. How do I compare these proportions and counts while taking ...
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How to fit a longitudinal model with binary outcomes

I'd like to fit a longitudinal model for where multiple subjects experience binary outcomes over time. To accomplish that, I'd like to use an additive random effect for each subject and an ...
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Power analysis for binomial data when sample sizes are different

I'm running an A/B test where I want to compare two different layout of a button and want to see if one layout lead to a higher Click-Through Rate with a certain statistical power. I want to use a ...
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How many answers to memorize for a test?

Recently one of my friends asked me this deceivingly simple question: I have a midterm with a predetermined list of 15 possible questions. Out of the 15 questions only 7 will actually appear on ...
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sum and product rules of probability

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
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1answer
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Caculate Type II error for Binomial distribution

Basically this is a question if I got the general idea behind the Type II Error right. I’ll set up a little reproducible example: Given $p_0=0.6$ with an $\alpha$-level of $\alpha=0.1$. As Binomial ...
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1answer
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Confidence interval for N, based on binomial

I've searched the web and these forums but couldn't find a subject that resembled my problem. If I missed something then I'd like to apologize upfront. My problem is this. I have a given probability ...
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1answer
50 views

What is a cumulative Binomial probability?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a cumulative Binomial probability is. So my question is: What are ...
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Test for aggregation of binary events/successes (binomial/glm??)

this has been vexing me for a while and I can't seem to solidify an answer beyond vague thoughts about Poisson distributions. I think this is a simple problem and I'm missing something obvious. Any ...
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How to apply the random statement in my model?

i am trying to model the following data in SAS using proc glimmix and i would like feedback if i have modelled my data correctly. My data consist of individual chickens (ID) who are grouped by ...
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1answer
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Generalized Binomial Models

With the standard Binomial probability distribution we consider n trials each with a probability of success p. This can be somewhat generalized to the Beta-Binomial Distribution which is effectively ...
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1answer
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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 ...
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Binomial probabilities in games when players can lie

An oracle can be asked a binary question. If the real answer is true, the oracle says true with probability tt (and false with probability tf=1-tt). If the real answer is false, the oracle says ...
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1answer
43 views

Finding the uncertainty in a Binomial probability estimate

In order to make some predictions for my work I have modeled a process using a binomial distribution, but in my case every single experiment must be a success and I am just changing the probability ...
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185 views

Probability mass function of the sample mean

Suppose that $X_1,X_2,\ldots,X_n$ is a random sample from a distribution with probability function: $$p_X(x)=\begin{cases} 1/2 &x=-1,1 \\ 0 & \text{otherwise} \end{cases}$$ Now if we define ...
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Variance of sample mean in a binomial distribution

I'm trying to derive that the variance of phat = pq/n. I'm unfortunately not getting that. Please tell me where I'm going wrong... ...
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1answer
37 views

Inverse Binomial Expected value

Let's $x$ be a random variable with a binomial distribution ($x \sim B(n,p)$). I know that the expected value of a binomial is $E(x) = n \cdot p $ but the inverse of a binomial? ...
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Binomial-like distribution

Let's say that $F(k; n, p) = \Pr(k \leq X)$ is the cumulative binomial distribution for $X \sim B(n, p)$. I am trying to solve for the smallest $i$ such that $F(i; n + i, p) > \alpha$ for some ...
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What is the most likely meaning of a confidence interval?

I'm using a service (Optimizely, as it happens), which show a confidence interval but doesn't explain what it is. I'm assuming it is a Binomial Confidence Interval, but I have no idea what the % ...
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22 views

Testing if a difference between two frequencies is different from zero

I have been trying to find the best way to test if the difference between two frequencies is significantly different from zero I have the following data: ...
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45 views

How do you verify a rating system?

Let's say I have a database of a lot of (thousands) chess games. In chess, every player have a rating, called ELO that gives an estimation how strong is this player. If we know the the ELO ranking of ...
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32 views

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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1answer
35 views

Why does a binomial glm give negative predictions?

I'm using count data in quite a simple way, but I cannot understand how a binomial glm can return negative predictions example code, where count of successes increases with responce variable: ...
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Confidence interval for a proportion estimated through stratified sampling

When estimating the confidence interval for a binomial proportion $p$, with $\hat{p}$ near 0 or 1, one has to use something other than the Wald interval to get a reasonable estimate (see for instance ...
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1answer
46 views

Dice probability - amount of “hits” in a pool

What I want is to calculate the probability of a minimum number of rolls with a minimum value for at given number of dice, e.g. "if I roll 20 dice, what is the probability of getting at least 3 dice ...
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2answers
75 views

Kruskal-Wallis?

I have three sets of data (1 set before treatment, second during treatment and the third after), each of which consists of hundreds of trials on a specific task that involves choosing 1 from 7 ...
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1answer
59 views

Logistic regression: can weights be used as a predictor variable?

I counted the number of birds in a flock, which gave counts like these: ...
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1answer
58 views

Standard deviation exceeds maximum value

I am trying to plot data that only has two possible values – 1 (present) or 0 (absent) – clearly not a normal distribution. The ...
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Binomial hierarchical model

I want to analyse the following data: 12 populations have been followed and a certain illness in the population is measured. I am interested in the probability of being ill. Now I divide these ...
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Sample size required for bivariate normal distribution

I am new here. Would like to ask a question on the sample size requirement for hypothesis testing following. If I am drawing samples with 2 non correlated variables (say x,y) from a bivariate normal ...
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Should I use a beta-binomial or binomial glmm?

I have several data sets on wildlife disease incidence. One of the issues with my dependent variable is that it represents only current infection status, therefore 0 (no disease) can represent either ...
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156 views

25 percent chance of winning, but only 3 wins in the past 20 events

If anyone is interested, this question is prompted by the recent NBA Lottery results. In the past 20 years, the team with the worst record in the NBA has won the lottery exactly 3 times. The team with ...
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37 views

Error for combining multiple binomial distributions

This problem is somewhat involved and I have a partial solution so bear with me. I will illustrate the problem with an example. Lets say we have two processes and we want to know which has a higher ...
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2answers
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Distribution for percentage data

I have a question about the correct distribution to use for creating a model with my data. I conducted a forest inventory with 50 plots, each plot measures 20m × 50m. For each plot, I estimated ...
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Statistically significant with binomial

What does this statement mean: "we calculated binomial 95% confidence intervals to determine the number of weeks that were statistically-significantly outside of the bounds of error"? How can one ...
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Testing classifier with binomial test when group sizes are unequal

I have data from 50 human subjects, who are divided into groups A and B (30 participants are in group A and 20 participants in group B). I also have a range of measurements from each subject. I have ...
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Appropriate GLM when response variable is proportion, but not binomial

The response variable I'm dealing with is the proportion of a total area that is suitable habitat for a species of interest. So although the response variable is bounded between 0 and 1, my intuition ...
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1answer
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Probability and Dice

Suppose that I roll a six sided die $x$ times, say 10, what is the probability that I will get a particular side/number, say 6, a specific number of times? Is there a general formula to compute this? ...
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Significance test of differences between two groups

I have two groups in which members have a property that is either true or false, i.e., each member either contains a gene or it doesn't. I want to test if there is a statistically significant ...
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Why is this distribution uniform?

We are investigating Bayesian statistical testing, and come across an odd (to me atleast) phenomenon. Consider the following case: we are interested in measuring which population, A or B, has a ...
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A binomial random number generating algorithm that works when n*p is very small

I need to generate binomial random numbers: For example, consider binomial random numbers. A binomial random number is the number of heads in N tosses of a coin with probability p of a heads ...
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Binomial testing for mutation patterns

I am looking for a certain type of mutation (DNA base A going to DNA base B when it is followed by C) in a dataset of 24 samples. I've noticed that this has happened 3 times out of the 4 times a gene ...
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1answer
48 views

Non-binomial posteriors for a binomial prior?

Let's assume we have a discrete binary random variable K (K=0 or K=1) for which the prior distribution is binomial. My understanding of Bayesian statistics tells me that regardless of the likelihood, ...
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Power when using binomial data versus continuous data

I have a question about something that is probably very basic to statistics but I feel I don't fully understand I've found that generally it is much harder to get high power (smaller) confidence ...
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How to specify a Bayesian binomial model with shrinkage to the population?

Problem I’m currently working on a problem where I have count data for $n$ items in the following form: ...
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Combining confidence intervals for a particular equation

I perform a Bernoulli experiment to obtain a binomially distributed probability $p_1$ with a 95% confidence interval $\delta p_1$. I perform a second, independent Bernoulli experiment to obtain a new ...
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1answer
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Course of Action for 2x2 tables with 0's in cell and low cell counts

I'm comparing two groups A, B after cross tabulating them with cases and controls. I get a table as such: Control Cases A 8 0 B 14 0 ...
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How to interpret results if a reference category of a categorical variable in multivariable logistic regression is not significant?

I am trying to do a multivariable logistic regression and using a normal binomial logistic regression, using binomial variable X (coded ...
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Three binomial random variables: test null that no two have equal mean

I have a dummy variable for health (0 or 1), and for each observation a treatment (one of three possible treatments for each observation). I want to test the null hypothesis that the proportion (mean) ...
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Sample size determination contradiction

I'm struggling with a contradiction when trying to determine a sample size when proportion is small. If I use binomial formula: $n = p(1-p)z^2/ME^2$ then the smaller the proportion - the smaller ...