Questions tagged [binomial-distribution]

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)

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41 views

Cumulative Multinomial Distribution does not seem to add up to 1.00 for parameters >3

For a multinomial distribution $P = ( n! / (\prod_{i=1}^k {n_i!}) ) * ( \prod_{i=1}^k {{p_i}^{n_i}})$ If you sum the resulting multinomial distribution for every possible unique frequency for k=2 for ...
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21 views

Upper Bound on the Probability of the Difference of Binomial Distributions

We start by defining a binomial difference distribution Let $X\sim \text{Bin}(n, p)$, $Y\sim \text{Bin}(n, q)$, $Z=X-Y$ I've found out that this distribution is somewhat difficult to write down ...
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How would I use a normal distribution approximation to perform a hypothesis test? [closed]

A company produces wine glasses in large quantities. It is known from previous records that 1% of the glasses have a hairline fracture. In a separate experiment, a random sample of 300 glasses were ...
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Distribution of $\sqrt{npq}(X-np)$ for large n where $X$ has a binomial distribution with parameters $n,p$ and $q=1-p$

Question: Let $X$ be a binomial random variable with parameters $n,p$. Let $q=1-p$. What is the limiting distribution of $a(X+b)$ for large $n$, where $a=\sqrt{npq}$ and $b=-np$? My attempt: The ...
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Central Limit Theorem - Asymptotic approximation to sums of Poisson random variables and Binomial random variables

Question: Let $X_1, X_2, ...$ be a sequence of iid $\textrm{Poi}(5)$ variables. Let $Y_1, Y_2, ...$ be a sequence of iid $\textrm{Bin}(10,\dfrac{1}{2})$ variables. Define the sums $S_n = \sum_1^n X_i$ ...
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Binomial distribution and sum

I need to solve this problem: Knowing that $n = 56$ and $p = 0.13$, and that X follow a binomial distribution, calculate $P_{x > 5} (X < 15)$. In other words, I suppose we have to calculate: $\...
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conditional expected number of success interpretation

Suppose $f(k,n,\rho)$ is the pmf for the binomial distribution, then $$\sum_{r=0}^nf(r,n,\rho)r$$ is the expected number of successes. My question is, how can I interpret $$\sum_{r=M}^nf(r,n,\rho)r?$$ ...
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1answer
26 views

Two consecutive zeros in Poisson random variables, and consecutive wins in a binomial setting

We have $X_1, ..., X_{100}$ i.i.d.r.v. having a Poisson distribution of parameter $\lambda$. I'd like to compute the expected value $E(N)$ for the number of times two consecutive $X_i$ are zero: $$N =...
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Name of this probability distribution

I want to name a process in which I have N = 100 individuals, and each independently has a probability p = 0.2 of an event occurring. The binomial distribution measures the probability of having n ...
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Binomial probabilities and limits

For $k =1,2,3, \dots,$ let $X_k$ be a binomial distribution with parameters $k$ and $0.5.$ Fix $0<a<1$ and assume that $a\neq 0.5.$ $\left\lfloor \cdot\right\rfloor$ denotes the greatest integer ...
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Confidence Interval for the product of heads and tails in 100 coin flips

If I flip 100 fair coins and then multiply the number of heads by the number of tails. Can you give a double-sided 95% confidence interval on the product of the number of heads by the number of tails? ...
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1answer
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Contradiction between me and my professor solution in normal app. to binomial

%$20$ of people smoke , if we choose a sample space consisting of $225$ people , what is the probability that more than $40$ people are smokers This is an probability question given by my professor....
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Why don't we reparametrize the Beta so its parameters represent number of heads and tails?

The Beta distribution with parameters $a$ and $b$ can be thought as the posterior distribution of the probability of heads when we start with a flat prior and observe $a-1$ heads and $b-1$ tails. And ...
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What is the name of the random variable $X$ with distribution $\mathbb P(X = i) = {n-i \choose k-1}/{n \choose k}$?

Fix integers $1 \le k \le n$ and let ${n \choose k}$ be the binomial coefficient. Question. What is the name of the random variable $X$ supported on $\{1,2,\ldots,n\}$ such that $\mathbb P(X = i) = {...
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32 views

probability winning game with changing winning conditions

I have the following problem: Suppose I have a 10 level game. You start at level 0, your direction is upwards (this will be of importance later). At each timestep you can ascend or descend a Level. ...
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1answer
51 views

Calculate the coefficient at $𝑥^3𝑦^4𝑧^5$ in decomposition $(𝑥 + 𝑦 − 𝑧)^{12}$ [closed]

Calculate the coefficient at $𝑥^3𝑦^4𝑧^5$ in decomposition $(𝑥 + 𝑦 − 𝑧)^{12}$ If I understood correctly, it needs to be decomposed by Binomial theorem. Thank you for writing your steps and ...
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Probability of probability confusion

I'm a little lost regarding the concept of estimators of probability in regard to the two following examples. Suppose I have $n$ marbles in a bag, and I sample them many many times and find that $20$ ...
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Eigenvectors of Stochastic Matrix

Let $N\in\mathbb{N}$. Define the matrix $Q\in\mathbb{R}^{(N+1)\times(N+1)}$ by $$Q_{ij}=\binom{N}{j}\Big(\frac{i}{N}\Big)^j\Big(1-\frac{i}{N}\Big)^{N-j}$$ for all $0\le i,j\le N$. The sum of each row ...
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A question about Central Limit Theorem and the calculation of a limit.

The question is the next: Using CLT to calculate $$\lim_{n \to \infty} \frac{8^n}{27^n}\sum_{k=0}^n\binom{3n}{k}\frac{1}{2^k}$$ I have started by defining a sequence $X_n$ of random variables with $...
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Calculate the following limit using the Central Limit Theorem

A question about the CLT Using Central Limit Theorem to show that $$\lim_{n \to \infty} \frac{8^n}{27^n} \sum_{k=0}^n \binom{3n}{k}\frac{1}{2^k}=0$$ I have tried to define a sequence ${X_n}$ with $...
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If $X$ is an observation from a $\text{Binom}(n, p)$, with $0 < p < 1$, how to find the UMVUE of $p^2$, and does it reach the Cramer-Rao lower bound?

If $X$ is an observation from a $\text{Binomial}(n, p)$, where $0 < p < 1$, how do I find the UMVUE of $p^2$, and does it reach the Cramer-Rao lower bound? I was able to find UMVUE of $p$. Since ...
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A randomly selected candle is lit for at least 3hr. What is the probability that it is original?

A pack has 4 original candles and 12 alternative. Their burning time is normally distributed with average and standard deviation: Original: X~N(4, 0.5). Alternative: Y~N(3, 0.25). Therefor: P(X)=4/16=...
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Binomial distribution exact

From a box containing 20 balls, 14 of them black and 6 white, we are drawing four balls with replacements. What is the probability, that black ball will be drawn twice? is my solution right? $n = 4, k ...
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How does one compute the total expectation of compund binomial experiments

Assume there are two binomial experiments. For example, we coin a toss n times and get k heads. If we perform second experiment k number of times (depends on first experiment), how can we compute ...
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How to evaluate Binomial(X,Y)? [closed]

Photo of Question For part c) of this probability question, I have never seen this kind of notation before and was wondering how the answer was evaluated? I understand that more than half of a group ...
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Poisson Distribution conditioned by Binomial Flag

I've this R code and I've to find the theoretical probability density function. ...
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Expected fraction of bins with exactly k balls, and finite bins

I need someone to challenge my logic. This problem concerns balls in finite-capacity bins. There are similar problems but all I can find relate to throwing balls into bins and discarding those balls ...
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Entropy of $Binomial(n, 1-2^{-1/n})$

I am interested in the entropy of the binomial random variable $Binom(n,1-2^{-1/n})$. Specifically, I would like to show that this is an increasing function in $n$.? I was able to verify this through ...
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Correctly find probability of binominal probability when $n$ and $p$ are sufficiently large

Let $X\sim Bin(30, 0.60)$ and $f(x)=x^2$ (A) Calculate the probablitiy P(X-1=30) and P(29>=X). My solution: Since $np=30*0.60=18>10$ and $n(1-p)=30(0.40)=12>10$, we can use the following ...
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Probability of majority votes being correct

Given a binomial distribution with $n$ expeiments, where probability of success is $p$. $\text{prob}(x=k) = \binom{n}{k} p^k (1-p)^{n-k}$ I'm trying to show that for odd values of $n\geq 3$, $p > ....
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Probability of the maximum of n binomial random variables being less than n

Suppose $x_1$, $x_2$, ..., $x_n$ are independent binomial random variables with parameters $p$ and $n$. Now what is the probability of $P[\text{max}(x_{1}, x_{2}, ..., x_{n})<n]$? I think we can ...
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1answer
60 views

Probability of that the number of heads in tossing a coin n times and n+1 times be equal [closed]

If we toss a fair coin ${n}$ times and another fair coin ${n+1}$ times; what is the probability of that the number of heads be equal?
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Does this discrete probability distribution have a name? (Related to Hypergeometric distribution)

Suppose there are $N$ objects, of which $K$ are of type 1 and $N-K$ of type 2. Objects of type 1 are indistinguishable and objects of type 2 are indistinguishable. One is interested in the probability ...
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1answer
46 views

Find marginal distribution of Y while knowing distribution of X and $Y|X$

Assume that X is uniformly distributed on (0, 1) and that the conditional distribution of Y given $X = x$ is a binomial distribution with parameters $(n, x)$. Then we say that Y has a binomial ...
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Can we prove that a fair coin tends to have 50% of relative frequency when number of trials tends to infinity ??

My first idea that was clear that wouldn't work is to try to prove that the limit of $${{n}\choose{n*p}}* p^{np}*(1-p)^{n - n*p}$$ tends to 1 when n goes to infinity. But it's obvious that it goes to ...
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The correct physical interpretation of Binomial distribution and bernoulli trial in this example

We know that every random variable can have a probability distribution. Examples include the number of heads in many tosses, or the number of ones on a dice after many rolls and so on. Suppose we use ...
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Expected value of sample entropy of a dice

A dice has $K$ sides, each side has probability $p_k$ of coming up in a roll, such that $p_k \geq 0$ and $\sum_{k=1}^K p_k = 1$. Let $X$ denote the random variable corresponding to such a probability ...
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Proof of the Third Central Moment of the Binomial Distribution without Moment Generating Function

$X$ is the sum of $n$ identical Bernoulli random variables, each with expected value $p$. In other words, $X_{1}, \ldots, X_{n}$ are identical (and independent) Bernoulli random variables with ...
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Lower bound on the probability for a Binomial random variable to be greater than another

Let $\epsilon > 0$. Let $(p_k),(q_k) , k \in \mathbb{N}$ be two sequences $ \in [1/2-\epsilon,1/2+\epsilon]$ s.t. $p_k>q_k$. For $k \in \mathbb{N}$, let $B_k(p_k)$ and $B_k(q_k)$ be two ...
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Is the classical definiton of probability correct, in case of probability distributions

Does the probability of success always depend upon the exact total number of favorable outcomes divided by the total number of outcomes, especially in the case of distributions? For example, consider ...
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51 views

Interpretation of mean in the following example

$\renewcommand{\Bg}[1]{ \Bigl( #1 \Bigr) }$Suppose I have a jar with $10$ blue balls and $20$ red balls. What I do is, I pick up the balls at random ( with replacement ) and find the probability of ...
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1answer
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Expectation of 3^x, where X~Binomial(10000,0.5)

Suppose that $X\sim \mathrm{Bin}(n,p)$, where $n = 10000$ and $p = 0.5$ What is $E(3^X)$? I know that $E(3^X) = 2^{10000}$, but I have no clue how to prove this. And in fact, through trial and error, ...
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Poisson parameter equal to product of binomial parameters

I'm wondering why the parameter $\lambda$ of a Poisson distribution ($Poisson(\lambda)$) is equal to the product between the parameters of a binomial distribution ($Bin(n, \, p)$): $$ \lambda = n \, p ...
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Playing tennis until one of the players wins 3 times (Binomial distribution)

Question Nadal and Federer playing tennis against each other. Nadal's probability to win one match is $\frac{2}{3}$, independently from the previous results. The two are playing until one of them wins ...
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we are trying to find most probable realization of random variable x, is this error in algebra?

So the task says: Random variable X has binomial distribution, what is the most probable realization of random variable X? so of course it must be p0 <= p1 <= ... <= pk and pk >= pk+1 >=...
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Why does the probability of an event change in a binomial experiment with the proportional change of successes and failures?

Let us assume the following two binomial experiments, assuming coin tosses with a fair coin $(p = 0.5)$: General: $\binom{n}{k}p^{k}(1-p)^{n-k}$ $\binom{10}{9} \cdot 0.5^{9} \cdot 0.5^{1} = 0.009766$ $...
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Problem formulating a question involving a binomial random variable.

I want to write a question which involves the use of a binomial random variable. This is it. "A student wants to pass a subject with 100 lessons studying the minimum possible. The exam is ...
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Explicit constants for a Chernoff-like bound

I recently came across the following bound (Lemma 2.1 of this paper) that is similar to Chernoff bounds: Let $\xi_i$ be i.i.d non-negative integer-valued random variables, with $E(\xi_i)=1$ and $Var(\...
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Show that $b(x;n,\theta) = B(x;n,\theta)-B(x-1;n,\theta)$

Show that $b(x;n,\theta) = B(x;n,\theta)-B(x-1;n,\theta)$ for $x = 0, 1, 2 ... n$ We have that $$B(x;n,\theta) = \sum_{k=0}^x\binom{n}{k}\theta^k(1-\theta)^{n-k} = \sum_{k=0}^x b(x;n,\theta)$$ What I ...
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1answer
48 views

Can we prove a trinomial distribution approaches a Gaussian

Consider a random walk with trinomial distribution where $i^{th}$ step is a random variable $X_i$ takes values $-\frac{1}{i}, 0, \frac{1}{i}$ with probability $0.3, 0.4, 0.3$ respectively. (A simpler ...

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