Articles of binomial distribution

Expectation (and concentration?) for $\min(X, n-X)$ when $X$ is a Binomial

I’d like to know if some results are known on the following type of random variables: for parameter $p\in[0,1]$ (for my purposes, $p < \frac{1}{2}$, and even $p \ll 1$) and $n \geq 1$, we let $X$ be a random variable following a Binomial$(n,p)$ distribution, and define $$Y \stackrel{\rm def}{=} \min(X, n-X).$$ Then, as a […]

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 the sum of the roll of two dices is a binomial distribution. Is this […]

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: I can understand most part of the proof except for this equation: I really don’t remember where it comes from, could anybody explain this […]

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) – \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My Question: Is there a better estimate for the constant $C$ than the one given above for the special case of the binomial distribution? Reason for my question: The given inequality for $C$ holds for any […]

How to prove that $\frac1{n\cdot 2^n}\sum\limits_{k=0}^{n}k^m\binom{n}{k}\to\frac{1}{2^m}$ when $n\to\infty$

I have solve following sum $$\sum_{k=0}^{n}k\binom{n}{k}=n2^{n-1}\Longrightarrow \dfrac{\displaystyle\sum_{k=0}^{n}k\binom{n}{k}}{n\cdot 2^n}\to\dfrac{1}{2},n\to\infty$$ $$\sum_{k=0}^{n}k^2\binom{n}{k}=n(n+1)2^{n-2}\Longrightarrow \dfrac{\displaystyle\sum_{k=0}^{n}k^2\binom{n}{k}}{n\cdot 2^n}\to\dfrac{1}{2^2},n\to\infty$$ $$\sum_{k=0}^{n}k^3\binom{n}{k}=2^{n-3}n^2(n+3)\Longrightarrow \dfrac{\displaystyle\sum_{k=0}^{n}k^3\binom{n}{k}}{n\cdot 2^n}\to\dfrac{1}{2^3},n\to\infty$$ so I conjecture the following: $$\dfrac{\displaystyle\sum_{k=0}^{n}k^m\binom{n}{k}}{n\cdot 2^n}\to\dfrac{1}{2^m},n\to\infty$$ $$\cdots\cdots$$ so I conjecture for $m$ be positive real number also hold.

Conditions needed to approximate a Binomial distribution using a Normal distribution?

Okay. So, many sources state different conditions for approximating binomial using normal. Variations I’ve seen are as follows. Assuming that we have Bin(n,p): 1. np and n(1-p) are both large (say, 20 or bigger) 2. large n, p close to 0.5 3. large n, np>9(1-p) My questions: 1. Are all 3 of these valid? Justify […]

The probability of a student speaking spanish is $30\%$. If we select $3$, what are the chances of at least one of them speaking Spanish?

In a school the probability of a student speaking Spanish is $30\%$. If we select $3$ random students, what are the chances of at least one of them speaking Spanish? So, I saw this question and tried to solve it, seemed like an easy question but I was wrong and still not sure why. Apparently […]

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 ?

Sharper Lower Bounds for Binomial/Chernoff Tails

The Wikipedia page for the Binomial Distribution states the following lower bound, which I suppose can also be generalized as a general Chernoff lower bound. $$\Pr(X \le k) \geq \frac{1}{(n+1)^2} \exp\left(-nD\left(\frac{k}{n}\left|\right|p\right)\right) \quad\quad\mbox{if }p<\frac{k}{n}<1$$ Clearly this is tight up to the $(n+1)^{-2}$ factor. However computationally it seems that $(n+1)^{-1}$ would be tight as well. Even (n+1)^{-.7} […]

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 $(n+1)p-1?$