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]
2302 questions
57
votes
4 answers
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 ?
54
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6 answers
Regarding a Coin Toss Experiment by Neil DeGrasse Tyson, and its validity
In one of his interviews, Clip Link, Neil DeGrasse Tyson discusses a coin toss experiment. It goes something like this:
Line up 1000 people, each given a coin, to be flipped simultaneously
Ask each one to flip if heads the person can continue
If…
Mystic Mickey
- 643
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2 answers
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
- 31,079
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votes
3 answers
A coin is tossed 100 times. How many instances of at least 5 heads in a row do we expect to see?
No overlaps. We are counting runs of at least 5, whereby for example a run of 6 does not count as 2 runs of 5.
I have received an answer to this question from someone with a PhD in Statistics, yet their theoretical answer does not agree with my code…
Stephen Taul
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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
- 7,852
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2 answers
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 the sum…
14
votes
1 answer
Looking for a limit
Looking for a limiting value: $$\lim_{K\to \infty } \, -\frac{x \sum _{j=0}^K x (a+1)^{-3 j} \left(-(1-a)^{3 j-3 K}\right)
\binom{K}{j} \exp \left(-\frac{1}{2} x^2 (a+1)^{-2 j} (1-a)^{2 j-2
K}\right)}{\sum _{j=0}^K (a+1)^{-j} (1-a)^{j-K}…
Nero
- 3,779
14
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2 answers
Why are all subset sizes equiprobable if elements are independently included with probability uniform over $[0,1]$?
A probability $p$ is chosen uniformly randomly from $[0,1]$, and then a subset of a set of $n$ elements is formed by including each element independently with probability $p$. In answering Probability of an event if r out of n events were true. I…
joriki
- 242,601
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2 answers
Asymptotic Expansion for $\frac1{2^n}\sum_{k=1}^n\frac1{\sqrt{k}}\binom{n}{k}$
Prior Art
The fact that
$$
\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
- 353,833
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votes
1 answer
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
- 2,640
12
votes
1 answer
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
- 1,027
11
votes
3 answers
Generalizing Poisson's binomial distribution to the multinomial case.
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…
shahensha
- 243
11
votes
1 answer
How to prove Poisson Distribution is the approximation of Binomial Distribution?
I was reading Introduction to Probability Models 11th Edition and saw this proof of why Poisson Distribution is the approximation of Binomial Distribution when n is large and p is small:
An important property of the Poisson random variable is that…
Demonedge
- 213
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7 answers
Combinations and Permutations in coin tossing
I understand the formulae for combinations and permutations and that for the binomial distribution. However, I'm confused about their application to coin tossing.
Consider three tosses. Outcomes with two heads are HHT, HTH and THH. So, there are…
Ian
- 133
10
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2 answers
Avoid unnecessary calculations when multiplying matrices if only need one element of resulting matrix
The Problem:
I need only the bottom left element of a product of matrices $(\bf{M_1}+\bf{I})(\bf{M_2}+\bf{I})\cdots(\bf{M_N}+\bf{I})$,
where $\bf{I}=\begin{pmatrix}
1 & 0\\
0 & 1\end{pmatrix}$, and…
WeCanDoItGuys
- 123