Probability of getting a seat in the train car

A train has got five train cars, each one with N seats. There are 150 passengers who randomly choose one of the cars. What is the probability that everyone will get a seat? I think that what is asking me is “what is the probability that each wagon is chosen by no more than N … Read more

Mixed model for three factors (split split plot design) with proportions as response

I am struggling with the analysis of a data set from an experiment done as follows: I have 5 blocks (that work as replicates) and each one of them was divided into two part (so split block), to each part a fertilization treatment was assigned (so fertilization treatment has two levels F and U). Each … Read more

test two binomial distributions: overall successes > failures and proportion successes A≠B

Let’s say I have data from an experiment: 3 participants took part each of them was subject to two conditions. Condition A, and B. in Condition A, 3 trials could yield success or failure (depending on the performance) in Condition B, a variable number of trials could yield success or failure (also depending on the … Read more

Compound Binomial distribution [closed]

Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? Add details and clarify the problem by editing this post. Closed 3 years ago. Improve this question I have come across the following expression: $$ T={m-y \choose \frac{m}{2}} \dfrac{1}{2}^{(m-y)}={m-y \choose \frac{m}{2}} \dfrac{1}{2}^{\frac{m}{2}}\dfrac{1}{2}^{\frac{m}{2}-y}$$ where $y$ is a binomial … Read more

Multinomial distribution with different probabilities for each trial

If in a binomial distribution, the Bernoulli trials are independent and have different success probabilities, then it is called Poisson Binomial Distribution. Such a question has been previously answered here and here. How can I do a similar analysis in the case of a multinomial distribution? For instance, if a $k$-sided die is thrown $n$ … Read more

Almost-Binomial Distribution (Different Success Chances per Trial)

Let’s say I have received complaints that my coin is not random enough after some people got 10 heads. My coin was truly random, but I want it to feel random, not be random. This is a pretty common problem actually. A lot of times people see patterns where no pattern exists. The solution is … Read more

What is the proper test for whether a particular option was chosen significantly more than two other options?

I have data in which respondents chose 1 of 3 options. I would like to test whether one of the responses was chosen significantly more frequently than the other two. If there were only two options I could use a one sample binomial test to see if it was chosen more than the constant of … Read more

Calculating confidence intervals in two sample analysis with extremely skewed count data

I want to identify the effect of a feature on a number of events. Each observation has >= 0 events and is assigned to one group, A or B. Each observation was assigned to a group by random sampling, and observations in group A were exposed to a treatment that group B was not exposed … Read more

Infimum of Binomial Confidence Interval Coverage Probability

Let X∼Binomial(n,p). Let ˆp=X/n be the estimator for p. Let the confidence interval for unknown p be ˆp±zα/2 √ˆp(1−ˆp)n. I am trying to find the infimum of the coverage probability for fixed n, where the infimum is taken over a subset of the range of (0,1). That is, I’m trying to find inf assuming 0<a<b<1. My … Read more

Binomial pp estimator

Let B(n,p) be a Binomial Distribution. We can estimate p using ˆp=∑mi=1Ximn, where Xi∼B(n,p) are i.d.d. and from Chebyshev’s inequality we have Pr since \mathbb{E}[\hat{p}]=p,\ Var[\hat{p}] = \frac{p(1-p)}{mn}. Therefore we can use O(1/(\epsilon^2 \delta)) samples and the above probability will be less that \delta. Question I would like to find a way to obtain an … Read more