In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. (Def: http://en.m.wikipedia.org/wiki/Binomial_distribution)

# Questions tagged [binomial-distribution]

1994 questions

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### Why is this coin-flipping probability problem unsolved?

You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars
the percentage of heads flipped. So if on the first trial you flip a head, you should stop and earn \$100
because you have 100% heads. If…

Joseph O'Rourke

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### Sum of two independent binomial variables

How can I formally prove that the sum of two independent binomial variables X and Y with same parameter p is also a binomial ?

Piyush Maheshwari

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### Intuition behind binomial variance

Suppose that I perform a stochastic task $n$ times (like tossing a coin) and that $p$ is the probability that one of the possible outcomes occurs. If $K$ is the stochastic variable that measures how many times this outcome occurred during the whole…

giobrach

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### Looking for a limit

Looking for a limiting value: $$\lim_{K\to \infty } \, -\frac{x \sum _{j=0}^K x (a+1)^{-3 j} \left(-(1-a)^{3 j-3 K}\right)
\binom{K}{j} \exp \left(-\frac{1}{2} x^2 (a+1)^{-2 j} (1-a)^{2 j-2
K}\right)}{\sum _{j=0}^K (a+1)^{-j} (1-a)^{j-K}…

Nero

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### Asymptotic Expansion for $\frac1{2^n}\sum_{k=1}^n\frac1{\sqrt{k}}\binom{n}{k}$

Prior Art
The fact that
$$
\lim_{n\to\infty}\frac1{2^n}\sum_{k=1}^n\frac1{\sqrt{k}}\binom{n}{k}=0\tag1
$$
is the topic of this question. An argument using a bit of probability theory gives a first order estimate of the size of the sum.
Estimate of…

robjohn

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### Why are all subset sizes equiprobable if elements are independently included with probability uniform over $[0,1]$?

A probability $p$ is chosen uniformly randomly from $[0,1]$, and then a subset of a set of $n$ elements is formed by including each element independently with probability $p$. In answering Probability of an event if r out of n events were true. I…

joriki

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### Finding mode in Binomial distribution

Suppose that $X$ has the Binomial distribution with parameters $n,p$ . How can I show that if $(n+1)p$ is integer then
$X$ has two mode that is $(n+1)p$ or $(n+1)p-1?$

hadisanji

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### Underdog leading at least once in an infinite series of games

We are observing a tournament where 2 players play a series of games. Exactly one player wins each game. So we can keep count and 5:3 might be the standing after 8 games.
The first player is the favorite and will win a game with probability $p > 1 -…

Niklas

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### Kullback-Leibler divergence of binomial distributions

Suppose $P \sim \mathrm{Bin}(n,p)$ and $Q \sim \mathrm{Bin}(n,q)$.
Their Kullback-Leibler divergence is defined by
$$D_{KL}(P||Q)=\mathbb{E}_{P}\left[\log\left(\frac{p(x)}{q(x)}\right)\right],$$
with $p(x)$ and $q(x)$ the pdf of $P$ and $Q$ resp.…

Hans M.A.

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### Distribution of weighted sum of Bernoulli RVs

Let $x_1,...,x_m$ be drawn from independent Bernoulli distributions with parameters $p_1,...,p_m$.
I'm interested in distribution of $t=\sum_i a_ix_i,~a_i\in \mathbb{R}$
$m$ is not large so I can not use central limit theorems.
I have the…

Alt

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### Binomial distribution: upper bound and lower bound

This question is discussed in Feller's Introduction to Probability Theory and Its Applications and i am relatively new in probability theory, so this theme is incomprehensible for me:
The probability at least $r$ successes is:
$$P(S_n \ge…

Daniel Yefimov

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### Why is the sum of the rolls of two dices a Binomial Distribution? What is defined as a success in this experiment?

I know that a Binomial Distribution, with parameters n and p, is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.
I read that when…

Stelios Avramidis

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### Binomial theorem from first principles?

I suppose I'll give a little context to this...
I was at first really excited by the prospect that both Bernoulli and Taylor's versions of $e^x$ actually ammount to the same thing (where, when you expand Bernoulli's definition…

fruitless fruit juice

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### Why is the Poisson Approximation to the Binomial Distribution Useful?

My textbook says that if $n$ (number of trials / independent Bernoulli random variables) is very large and $p$ (probability of success per trial / Bernoulli random variable) is very close to $0$ or $1$, we can approximate the mass associated with…

user865043

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### What is the distribution of all sums of numbers from the set $\{1,2,\cdots,n\}$?

I was wodering: if you have the set of integers $R = \{1, 2, \cdots , n\}$, I would like to know the distribution of the sum of the members of all the posible non-empty subsets. I have done a simple calculation for some values of $n$ and here you…

Carlos Toscano-Ochoa

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