Questions tagged [binomial-distribution]
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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Concentration inequality for hypergeometric distribution
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
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Testing the hypothesis that $\sum_i^N p_i^2 =(N-1)/2N$ given $\sum_i^N p_i=1$
In ecology there is a simple index for measuring diversity known as the Simpson index. In it simplest formula, it takes a series of proportions, representing the relative abundance (number of ...
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What exactly is likelihood? [duplicate]
My understanding about likelihood, given some reading, is that it is how likely we are to observe the actual data given a certain parameter or parameter values $\theta$. Like with the coin toss ...
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Joint and conditional probability with Poisson and Binomial distributions
I'm having a hard time trying to figuring out how to resolve a problem that involves a Poisson distribution and a Binomial distribution.
Let's suppose that the total number of offspring (sons and ...
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How many items do I need to inspect given my population size?
Hey just scratching my head with this problem:
If I have a process that produces say 10,000 items. Each item can either be correct or incorrect (pass or fail). How many items do I need to check to be ...
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Testing difference between datasets from binomial distributions
We will be conducting an experiment where a group of $N$ people take a test that has $M$ independent yes-no questions. We compute the number of questions each person $j$ gets correct as a score $m_j$. ...
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How to prove the general formula for variance of binomial distribution
In the video from Khan academym Khan tried to prove that the variance of the binormial distirbution follow the formula
$$ D(x)=np(1-p)$$
However, I am a bit confused about what he said in the video. ...
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Estimating number of occurrences of binomial tests
I have data representing a counting of the number of successes in a series of $n$-trial binomial experiments, however each experiment might have a different $n$ and is unknown.
So, if I for example ...
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Why are the p-values for a binomial test very different from the p-values for a chi-squared test?
I am testing if a coin is fair by throwing it n-times and having n/2 + sqrt(n) heads.
However, I get very different p-values ...
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Binomial regression in R: lm() with logit, vs glm() with family=binomial
I am currently learning about GLMs. Suppose my predictor variable generates a number of "successes" and "failures" as the response variable. Naturally I want to model the ...
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how to calculate p-value based on # of trials and observed values?
In this article, the p-value is calculated as 0.32 for observing 55 heads in 100 coin flips.
https://netflixtechblog.com/interpreting-a-b-test-results-false-positives-and-statistical-significance-...
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Applying filter (e.g., moving average) on Binomial distributed random values
I start with Binomial distributed random values with known $N$ and success probability $p$. I can easily estimate PMF and CDF of such distribution. Now assume these Binomial distributed random values ...
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Calculate $E[X]^2$ where $X \sim \operatorname{Binomial}(n,p)$ with binomial coefficients expansion [closed]
Calculation of $EX$ using the binomial expansion formula is easy:
\begin{align}
EX &= \sum_{x=0}^{n}x\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x}\\& = np \sum_{x=1}^{n}\frac{(n-1)!}{(n-x)!(x-1)!}p^{x-...
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Is it appropriate to present predicted probabilities from emmeans for a mixed-effects binomial logistic regression?
I am trying to understand how to analyze data for a generalized mixed model (GLMM) with a binary response. I am interested in visualizing the predicted probabilities, as well as a measure of effect ...
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Upper bounds on $\mathbb{P}[X \leq k]$ when $k > \mathbb{E}[X]$, for binomial rand. variable $X$
Let $X$ be a binomial random variable, $X \sim \mathcal{B}(n,p)$.
When $k > \mathbb{E}[X] = np$, are there no Hoeffding-like bounds on the probability $\mathbb{P}[X \leq k]$?
When $k \leq \mathbb{E}...
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Distribution of a single number in roullette but not chosen a priori
In roullette (say European with 37 slots) when the wheel is fair and the success is predefined as hitting a specific number (say $13$) the number of successful outcomes in $n$ spins/trials should ...
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Using a binomial distribution to mathematically quantify a thought experiment from my friend (python)
My friend asked me the following question:
...
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Bivariate problem, what is the best model for this? Scoring probability depending on distance
I have been directed here from the main stackoverflow site, where i described my problem: here.
Basically i have a set of sports data containing shots with their distance to the goal and the result (...
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Given that quasibinomial regression models extra-binomial variation, why ever do binomial regression if quasibinomial is more flexible?
In reading about quasibinomial regression:
The quasi-binomial distribution, while similar to the binomial distribution, has an extra parameter 𝜙
(limited to |𝜙|≤min{𝑝/𝑛,(1−𝑝)/𝑛}
) that attempts ...
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How to aggregate survival type of data to single measure independent of time?
Thank you for the opportunity to ask this question. Since I do not speak fluently (still learning) the language of mathematics, I explain this in plain English. I understand this is too ideal but I ...
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Binomial distribution with two probabilities
Here's the problem I have:
The probability that an apple is red is 80%.
The probability that a red apple is rotten is 10%.
If 10 apples are picked at random, what are the odds that half are red and ...
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How to construct confidence intervals for multiple choice survey responses where the measurement is the percentage of respondents?
Say I have $k$ categories of responses to a survey question from $n$ respondents. Assume n of each category is about 50, so we can't just work off asymptotic behavior necessarily.
How do I compute ...
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one-sided binomial confidence interval using brute force on CDF rather than Clopper-Pearson/Wilson/etc
I was recently trying to solve a marketing problem. For say 300 trial users, only 3 converted to paying subscribers. The conversion rate for this sample is obviously 1%. And almost as obvious, we ...
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Highly Skewed Binary Response Variable 10% One Category, 90% Other
I have a highly skewed binary response variable in my dataset. I am having issues finding a model that meets all assumptions for a binomial distribution GLM, and I believe this may be correlated to ...
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Bayesian Analysis of Coin Toss with Three Outcomes: Incorporating a Fixed Probability of a 'Side flip' event
I'm working on a Bayesian analysis of a coin-toss scenario and have a conceptual question to clarify my understanding.
Background
Given a uniform prior on the probability that a coin lands tails over $...
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Distribution model for Multiple-choice Data?
I am running an experiment where I am testing the effects of three interventions (A,B,C) and measuring participants' performance via a multiple choice test with 5 questions. To perform hypothesis ...
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Covariance between two binomial random variables or expectation of product of binomial random variables
I have an empirical distribution $S_n(x)$ (= proportion of samples less than equal to x) from a random sample $X_1, X_2, ..., X_n$ for a random variable $X \sim F_X$. Consider the random variable $T_n(...
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Distribution of the Sum of Dependent Bernoulli Random Variables
Suppose that we have two random variables consisting of an indicator function like below:
\begin{align*}
V_1=1[c_1\geq u_1] \\
V_2=1[c_2\geq u_2]
\end{align*}
where $c_1$ and $c_2$ are constants, and $...
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How to check assumptions of a binomal GLM with categorical predictors
I have a data set that looks like this (subset below):
...
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Will this converge to origin?
Suppose you have a diffusion of 100 points with the following iteration:
$$(x_{n+1},y_{n+1}) \sim \mathcal{N}\left((x_n,y_n), \frac{x_n^2 + y_n^2}{2} I_{2 \times2}\right)$$
This will make a high ...
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Sample size in Sample Proportions
I am in high school learning about sample proportions and they say that $n$ is the sample size. The example they gave is you spin a spinner board where the chance of landing on a 1 in 0.6, and 0 is 0....
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Estimate of the sample proportion with
If I have a company with 1000 employees and 20% are female, and I want to select 20 of them at random, can I say that there is 95% certainty that the number of female employees in the sample will be ...
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Testing the effect of a customer loyalty initiative
I work for a company with a chain of auto workshops. We are interested in testing the effect of a data gathering initiative on customer loyalty (visits per year) and spend per visit, hoping that the ...
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(Using conditional expectation to calculate) expected value of the product of two dependent random variables
Let $\mathbf{X}$ be Binomial point process in $W = [0, 6] \times [0, 4]$ with $n$ points. Let $A_1 = [0, 2] \times [0, 4]$, $A_2 = [0, 6] \times [0, 2]$, and $A_3 = [2, 6] × [2, 4]$. I want to find $E[...
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How do we obtain the posterior of a beta binomial mixture of continuous and a discrete density?
In section 3.6 of Jim Albert's 2009 book "Bayesian Computation with R" he describes a test of whether a coin is fair using a mixture of priors. The coin tossing follows a binomial ...
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glm with family=bernoulli vs glm with family=binomial
What is the difference in terms of the coefficients the two yielded in R; I suppose they estimate the same thing? (Log) odds ratio? Or is the latter estimating risk/relative risk ratio?
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Scaling variables for model selection. Unscaling for final model?
I struggle with the following question:
I had to scale variables for model selection as they were in very different units and the model struggled to converge. Now that I have a final model the ...
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Baysian Inference with beta-binomial model with different number of trials
Let's imagine that we have created and estimated a posterior with a beta prior and binomial likelihood function. The posterior is given by $p(\mu |D) = Beta(2,99)$. Now, I want to do inference with ...
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Combining two success runs in parallel
The goal is to calculate the reliability of a process. Here reliability is defined as follows:
Definitions and tests I used
Let $X$ be a random variable that is equal to $1$ when no defect is present ...
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Correct Distribution to Calculate Probability of Ballot Measure Success
The CA Secretary of State has historical records on ballot measures starting in 1912, and we're attempting to calculate probabilities for the 2024 ballot measures based on the historical average. Over ...
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Proportion data with dependency using binomial generalized linear model
I did an experiment on small tree branches with closed buds were a box containing 30 branches was monitored 3 times per week for the presence of leaves. The number of branches that have reached a ...
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multiple comparison with a binomial model in R
When performing a multiple comparison in a binomial model, I encounter a problem when one of the groups has the probability of success = 1. For example,
...
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$N \sim$ Poisson iff number of heads & tails independent?
Consider $N$ to be the total number of fair coin tosses such that $N \sim \mathrm{Pois}(\lambda) $.
Then for the events (sorry) random variables defined as, $A:$ No. of heads observed and $B:$ No. of ...
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How can I plot interactions of standardised coefficients from a binomial logistic regression?
I'm using the mvabund package to run a model of species presences as a function of a trait matirx and an environmental matrix. The '4th corner' terms are the ...
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Coding error? Why is the treatment not showing as a significant effect with GLM and standard error is so high?
I've been handed off some data to analyze and I'm looking for
something simple and straightforward. The main study question is
testing the efficacy of 7 herbicides, with one untreated control. The
1 ...
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Calculating probability curve of a binomial sample
Apologies, I'm not a stats-person, I'm looking for a concept that I'm struggling to Google.
I have a boolean function that returns true or false at some unknown chance c. I have a sample of 100 ...
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How to generate binomial point estimate consistent with a binomial confidence interval?
Consider this sample data:
...
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Binomial vs product of binomial in likelihood for Bayesian inference
I am working through McElreath's book on statistical rethinking. One of the problems is the following:
Using grid approximation, compute the posterior distribution for the probability of a birth being ...
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Understanding the Difference between Bernoulli Distribution and Binomial Distribution
In my recent study, I conducted 67 measurements and recorded 11 successful outcomes. Now, I am seeking clarification on the appropriate formulas to calculate the measurement error. Should I use the ...
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Probability of two sum and alternating sum of two random variables being of opposite signs
Consider the random variables
$a_i,i=0,1,\ldots,n$ be random variables which take values from $\{-1,1\}$ independently and randomly with equal probability. Let
\begin{align}
S &= a_1+\cdots+a_n , ...
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