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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Z-test for proportions with unequal variance and unequal sample sizes

I need to test a null hypothesis that two proportions are drawn from the same population. Data ...
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Binomial Bayesian Regression with Metropolis Hastings in R

I'm trying to implement the Metropolis Hastings algorithm in this problem but I'm having problems with the convergence. $$Y_i|\beta_0,\beta_1 \sim \text{Binomial}(m_i,\theta_i)$$ where $logit(\theta) =...
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Poisson versus Binomial and estimation of variance [duplicate]

Let X denotes the number of successes in n trials and p is the probability of success. Then, X ~ B(n, p) and EX = np, VarX = npq. If n is really large (e.g. n > 20000) then it seems also reasonable ...
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Sample and rbinom generating disparate results?

I am sure I am just misunderstanding this but rbinom and sample seem to be generating disparate results and I am not sure why. I am trying to create a categorical factor with a balanced third in each ...
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Comparison of total counts vs. comparison of mean counts per participant in exploratory data analysis

I already came across this question: Difference in means vs difference in proportions, but could not answer my question based on the answers and comments there. My question would be this: Suppose ...
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what is the distribution of Proportion?

I have some observations are recorded as Proportion, from a binomial distribution with unknown success numbers. Typically in standard case, we observe only number of successes say $X$ out of trials ...
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Calculate probability of minimum number of sample types within a sample distribution - z-statistics vs. binomial distribution

(The title may have to be changed, because I am learning statistics and can't describe problems through the use of the correct terminologies yet) I have the following question: "90% of all plant ...
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Accumulation of evidence in basic statistics

Imagine I have a coin, and I want to study tossing outcomes of that coin. I do this over two experiments. In the first experiment, I toss the coin $n_1$ times and write down all the outcomes, in the ...
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alternative to ANOVA or simplifying approximations for multi-factor,, batch design with binomial data (advanced beginner)

this is my first post. I did already read various didactic materials and related posts here, but still had trouble figuring this one out. Sorry this is so long but I am trying to be as specific as ...
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CIs for random intercepts from gam and stan_gamm4 are similar, but those from gamm4 are not. why?

Question: how can one obtain a good confidence interval for the estimated random effects from gamm4? Motivation: The example below, using binomial data with a random intercept, shows that estimated ...
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How does the following term happen when using the Wald Test on two-sample binomials?

I'm having trouble understanding one of the steps of the Wald test used in two-sample binomial proportions. More specifically, when $X_1, ..., X_{n_1}$ are iid Binomial$(1, p_1$), $Y_1, ..., Y_{n_2}$ ...
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Confidence interval for a rate ratio

My question is easy, but I really got stuck here ... Say I have 500 people, among them, 200 likes eating ice cream. Among these 200, 60 likes strawberry ice cream. So the rate ratio of ice cream vs ...
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h1b lottery procedure change to increase proportion of masters. Can someone explain how the expected proportion is different in the new procedure?

Simply speaking, there are 85,000 spots. 20,000 is reserved for masters(or higher education). Previously, they were selecting 20k first from everyone with a masters and then grouping everyone ...
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GLMMadaptive for binomial repeative data

am doing repeated binomial data with GLMMadaptive in R. Outcomes were collected at 3 different time points(collected at baseline, 12 weeks, 24 weeks)and data looks like: I want to calculate the ...
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Using conjugate priors to estimate the posterior distribution of a proportion of a region composed by subregions

Let's say I have a region that is divided into 3 subregions. In each subregion, I run ~90-110 randomly allocated surveys asking a binary question. I want to know if the way that I am estimating the ...
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Urn problem (draw two balls of different colours with replacement)

e.g. 3 white balls and 7 black balls. Prove that the probability to draw two different coloured balls is 0.42 I get the intuition, but can't formally explain the probability.
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Consecutive coin flips, what is the appropriate statistical test for this word problem? [closed]

I was listening to a podcast by NDGT (Neil deGrasse Tyson, a prominent scientist) and he posed a simple thought experiment to illustrate the susceptibilities to cognitive bias. What I've come here to ...
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Computing sample size for a sample to estimate binomial distribution when point estimate of proportion is 0

For example, I have a estimate of p = 0 (95% CI: 0, 0.01), and I want to know the sample size for this sample. For $p\not =0$, I can compute the s.e. by $\sqrt(p(1-p)/n)$ to get the $n$, but not sure ...
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Number of confidence intervals for Binomial Distribution

Consider the random variable $X$ to denote the event that the confidence interval cover the unknown parameter or not. Thus, X will follow the Bernoulli distribution with a parameter $θ$, i.e. $X~Be(θ)$...
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Bounds on the conditional variance of a truncated binomial

I have a binomial variable $R$ drawn from $binom(N, p)$, and I'm interested in the variance of $R$, given $R \ge Q$. The pmf of this variable $R^*$ is $$ \phi_{R^*}(l) = \frac{\phi(l, N, p)}{P} $$ ...
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Testing sequence of ones and zeros for randomness

I am given a sequence of $40$ ones and zeros and I have to test the null hypothesis that ${40 \choose n_1}$ sequences are all equally probable ($n_1$ being the number of ones). To do so, I have to use ...
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How to deal with ragged power curve

I am exploring various methods for sample size calculation and power analysis from binomial data, and frequently coming up against ragged power curves that go up and down as sample size increases. I ...
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Is it possible to prove that this function of the truncated binomial is decreasing?

Suppose a random variable is binomially distributed with $G$ trials and success probability $p$. Consider the expected proportion of successes, given that the proportion of successes is at least $k$: $...
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Difference between geometric distribution expectation and 1 - failure with Binomial

I'm trying to understand a simple problem: How many times you'd need to roll two dice to get two ones in a single roll. One way I see this is as a problem the geometric distribution describes. You ...
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How can I determine which of two sequences of coin flips is real and which is fake?

This is an interesting problem I came across. I'm attempting to write a Python program to get a solution to it; however, I'm not sure how to proceed. So far, I know that I would expect the counts of ...
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Lower confidence interval for quantile function [duplicate]

I have a real-valued, unknown distribution $\mu$ and would like to find the largest threshold $t \in \mathbb{R}$ such that $\Pr_{X \sim \mu}\left[X \leq t\right] \leq q$ with high probability $1-\...
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Coeficients from binomial probit model with more than one predictor

A bird observed at time t will have started moult when its time of onset is before t. This occurs with probability given by Prob (started moult before time t) = Φ[(t − μ)/σ], where the function Φ ...
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How to estimate upper CI of population prevalence of (very) rare events - with weights

I'm doing something equivalent to estimating the prevalence of a rare disease. I might sample 1000 individuals, and find zero cases. In which case I'd calculate the upper confidence interval using ...
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Is there a test (in R) for predicting the proportion of "successes" to "failures"?

As I understand it, binomial logistic regression is good for predicting the probability of a "1" outcome from a discrete binomial distribution. What if instead, one wanted to predict the ...
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Goodness of fit for multinomials with small np

I've got a model with continuous explanatory variable $x_i$ and some positive integer $m_i$ for each datapoint, and then a multinomial response - i.e. $p$ categories, and the sum of counts across the $...
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Probability of outcomes from overlapping samples of a random variable

I have a feeling there is a straight forward answer to my question, but I'm not sure what the appropriate term to search for is. I'm trying to get a sense of what the probability is of finding some ...
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Binomial mgcv::GAM with different behavioral states - how to interpret results

I'm trying to understand whether I have diel patterns of behavioral states of sampled animals. Per hour, each animal is assigned one of three states based on how much time it's spending in each state: ...
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Find the probability of which sample comes from a "higher" distribution based on random sample from two distributions

I am trying to find the answer for the following question. I have two distributions of numbers between 1 and 8, as illustrated here: I draw random samples from each distribution, which we can call &...
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Fisher's Exact Test for 4 subgroups

I have a question for using Fisher's exact test for subgroups. I have 39 subjects that was divided into 4 groups( n = 9, 5, 10, 15, respectively). There were 3 conditions dividing the groups. Table ...
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Choosing statistical test for Preference-shift

I am having trouble choosing the proper method to support my hypothesis. I have conducted an animal study, where subjects chose A or B, 6 times a day for 3 days (day 1~3). At first, I divided the ...
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Binomial test with Chance level prediction

Thanks for reading. I wanted to ask if my logic I used for my experiment makes any sense. My hypothesis is that watching Pepsi commercial increases Pepsi preference. My experiment has 2 steps. (1) ...
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Distribution for Fraction of Success in a Binomial Setting

So the actual original question I am trying to solve is a little bit different than the title: In a binomial setting with probability of success $p$, I keep examining observations until a fraction of ...
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Setting null hypothesis for Binomial test

I have a question for setting null hypothesis in binomial test. More specifically, is there any way to assume chance level when it is unknown? For example, let's say that I have a slightly bent coin. ...
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binomial distribution with changing probability of success across samples

I've got a scenario question of: An economist interviews 75 students in a class of 100 to estimate the proportion of students who expect to obtain a ‘‘C’’ or better in the course. Is this a binomial ...
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Expectation of indicator variable squared to compute variance of a binomial

$$ X \tilde{} Bin(n,p)$$ I know that X can be written as the sum of indicator variables like this: $$Then ~ X = I_{1} + ...+ I_{n}, where,~ I_{i} = 1 ~ if~success,~0~ otherwise.$$ This is quite ...
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If $X_1 \sim \text{binom}(p_1,n_1)$ and $X_2 \sim \text{binom}(p_2,n_2)$, how to prove that the MLE of $p = p_1 - p_2$ is $\hat{p}_1 - \hat{p}_2$?

Suppose $X_1 \sim \text{binom}(p_1,n_1)$ and $X_2 \sim \text{binom}(p_2,n_2)$, where $X_1$ and $X_2$ are independent, and let $p = p_1 - p_2$. How can I prove that $\hat{p} = \hat{p}_1 - \hat{p}_2$? (...
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Different types of residuals revealing different stories on model fit

I have a series of observations as y_actual and fit a glm model with binomial family using <...
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How to test if watching a advertisement only 1 time is enough to change one's preference

I'm sort of having a trouble finding what statistical test to use in a specific project of mine. My experiment was as follows: I have 10 friends and they all prefer Coke cola then Pepsi. Each of them ...
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Cohens d in relation to a single case

Cohen's d in relation to a single case The average effect size (Cohens d) for insomnia-treatments ranges from about 0.3 to 0.4. If I had a patient that suffered from insomnia 4 out of 5 weekdays (...
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Blocking Binomial Data (Bernoulli Trials)

Imagine you have a colony of roaches and you want to compare the efficacies of two insecticides. On one day you apply insecticide A to 50 insects and record your "outcome" as "1" (...
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Convergence algorithm for power analysis?

I found this algorithm on a shared drive and I'm really curious how it works. From appearances, it's used to estimate the MDE in power analysis when the effect size (difference of proportion means) ...
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Mean and Variance of an estimated squared proportion from a Binomial distributed random Variable

I want to make a confidence interval on the number of successes $K$, given a sample number $n_2$ and a proportion $h$, where the $K$ would be Binomial distributed. $$ K \sim Bin(n_2, h) $$ However, ...
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Probability distribution of binomial of binomial divided by same binomial

Let $Y \sim \mathrm{Bin}(n, q)$ and $X \sim \mathrm{Bin}(Y, p)$, is there an expression for the distribution of $X/Y$?
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Sample proportion vs mean proportion for sample size estimation

Context I would like to estimate the sample size needed for an experiment. I’m testing a feature on a website and would like to detect a significant change between different variants . One control and ...
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Is the likelihood of a discrete binomial variable same as it probability? Like in the case of tossing a coin lets say 12 times

I am working on a probability project where we first generate random variables for a given Binomial experiment and then we generate a PMF for 10 coin tosses using the list of random variables we ...
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