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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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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3 votes
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Proof that $g(p)$ unbiasedly estimable only if it is a polynomial (Binomial Distribution)

In Lehmann-Casella (Theory of Point Estimation) they state without proof that if $T \sim Bin(n,p)$, then $g(p)$ is estimable only if it is a polynomial in $p$ of degree $\leq n$. How does one go about ...
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how does adding information change the probability of an event?

In a comedy TV programme, four men are sitting at the bar. The barman tells them: "Did you know that, statistically, one out of four men is having an affair?". The first man replies "...
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2 votes
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Calculating confidence interval for binomial distribution [duplicate]

Suppose we have a sample $X_1, X_2, \ldots, X_n \stackrel{\text{iid}}{\sim} Binomial(\theta)$, where $n$ is known to be large. I would like to calculate the 95% confidence interval for $\theta$, and I ...
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Weighted sum of Bernoulli distributions

Suppose $$ X_i \sim \text{Bernoulli }(p) $$ What can be said about the distribution of $Y$, given by $$ Y = \sum_{i=1}^N w_i X_i $$ for non-negative weights $w_i$? Note that $N$ here is small, so ...
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coefficients for small effects in a binomial simulation

Im trying to simulate a binomial linear model, where a binary output is predicted from a three level categorical variable. Supposing that 1.In the base level, the event has a small (.15) probability ...
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Proportion data with number of trials known (and separation?): GLM or beta regression?

I perform a lot of bioassays in which I score mortality not on individuals, but on groups of individuals as a proportion (the denominator, i.e., number of trials, is known): Sample Data: ...
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distribution r studio

About 12% of males are colorblind. A researcher needs three colorblind men for an experiment and begins checking potential subjects. What is the probability that she finds exactly 3 or 4 men in the ...
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1 answer
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Variance of a sample

If the outcome of a data generating process is always a 0 or 1, is the variance of the sample npq? Or is this variance formula only applicable when the data generating process is as simple as 'draws ...
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Valid Binomial Test

I'd like to understand if I'm appropriately using a binomial test on my dataset. I have users who can perform an action, tracked by day of the week. For each user, I'd like to automatically detect ...
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1 vote
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Computing the power for binomial regression with indicator variables

My question is kind of a sequel to this one. For an experiment I'm designing, I want to model the outcomes by a binomial regression, something like ...
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Is an unbalanced categorical independent variable a problem for binomial logistic regression? (elaboration on a previous question)

I'm trying to predict the probability of an event to lead to a yes or no response. The independent variable is a categorical predictor with 3 levels. I tested 21 infants; their responses to 3 ...
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4 votes
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Is an unbalanced categorical independent variable a problem for binomial logistic regression?

My dependent variable is a yes/no measure; I'm trying to predict the probability of an event to lead to a yes or no. The independent variable is a categorical predictor with 3 levels. My data is ...
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2 votes
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Does joint probability apply when calculating likelihood with the binomial distribution?

I am trying to understand this picture from Statquest in the light of the Wikipedia statement that Likelihood describes the joint probability of the observed data as a function of the parameters of ...
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How to find the MLE for $\theta$ in terms of K and n

this is my first question here :) This was seen in a machine learning exam: Suppose that $X_1 ...X_n$ are n i.i.d random variables with the following distribution: $f(x;\theta) = \begin{cases} \...
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1 vote
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AIC for robust generalized linear models (glm)

How can I calculate Akaike's 'An Information Criterion' with small sample size correction (AICc) for glmrob from robustbase in R?...
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Can we use a GLM(M) to compare binomial data to an expected distribution?

We conducted an experiment with 19 birds together in an aviary (11 of them were infected, 8 healthy). We released hundreds of mosquitoes for a night in the aviary, collected the blood-fed mosquitoes ...
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