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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Computing the variance of $X \wedge (n-X)$ where $X\sim \text{Binomial}(n,p)$

I want to calculate variance of $Y=X \wedge (n-X)$ where $X\sim \text{Binomial}(n,p)$ and $n$ is even. I have managed to determine the distribution of $X \wedge (n-X)$. We have $$\mathbb{P}(Y=k) = 2\...
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Inverse Mellin transform of the Mellin transform of the binomial PMF

The probability mass function of the Binomial distribution is given by $$\begin{equation} f(x)=\binom{n}{x} p^x (1-p)^{n-x}, \end{equation}$$ where $p \in [0,1]$ and $x=\{0,1,\dots,n\}$ (finite ...
Efficiency's user avatar
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Sum of binomial coefficients weighted by 1/t

I would like to evaluate the following sum in terms of $n$. The sum is essentially a weighted sum of a binomial distribution with N = 2n and p = $\frac{1}{n}$. I can't figure out how to do it, despite ...
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Binomial distributions to find unknown probability [closed]

The Sordaria is a small fungus containing many spores arranged in sets of eight spores. As each set is ejected from the organism, the bonds joining the spores may or may not break. Data are collected ...
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Convergence in Exponential Scale: Binomial Distribution with Exponentially Many Terms and Small Success Probability

Assume that the dot equality notation $a_n \doteq b_n$ denotes equality in the exponential scale for two positive sequences $\left\{ a_n\right\}$ and $\left\{b_n\right\}$, implying that the limit as $...
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Limit probability of a side of a dice showing up more than $p_i + \omega(\sqrt{n})$ times

Consider a dice $D$ with sides $\{1,\ldots,6\}$ and let $p_i$ be the (constant) probability of each side). Let further be $A_n$ the event that any side shows up more than $p_i + h(n)$ for $h(n) = \...
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What is the PMF of the ratio of two dependent sums of weighted Bernoulli random variables?

Let $ X_i $ be a Bernoulli random variable of success probability $p$. Is there any closed-form expression of the probability mass function of the following quantity: $$ \frac{\sum_i a_i X_i + b}{\...
Kevin B.'s user avatar
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Binomial distribution with exponentially many terms and exponentially small success probability

Consider the following Binomial distribution, $$ N \sim \operatorname{Binomial}(e^{nA}, e^{-nB}), $$ In other words, it's a Binomial distribution with exponentially many terms and exponentially small ...
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Prove that for all $k \in \mathbb{N}^*$, $\mathbb{P}(S \ge k) \le \dfrac{1}{2^n}\left(\dfrac{en}{k}\right)^k$

Problem: Let $b$ be a random variable such that $\mathbb{P}(b=0) = \mathbb{P}(b=1) = \dfrac{1}{2}$. Let $b_1,\ldots,b_n$ be independent copies of $b$ and $S = \sum_{i=1}^n b_i$. Prove that $$\forall k ...
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Best strategy for choosing valued items. (variant of secretary problem) [closed]

Suppose we have to fill our bag with items and each item has a value $V_i \sim \text{Uniform}[a,b]$ where each random variable is iid. There are $N$ items in total but only a fraction $x$ of all items ...
Eduard's user avatar
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Why are the Binomial and Poisson probabilities different for frequency of a successful dice roll?

Why do I get different results using a Binomial vs Poisson process for calculating the frequency of a successful roll of a 6 sided die? Should I get the same answer calculating the same scenario with ...
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How to differ between Binomial and Hypergeometric Distributions while solving problems?

I'm listing 3 questions: 1.Suppose that a batch of 100 items contains 6 that are defective and 94 that are not defective. If X is the number of defective items in a randomly drawn sample of 10 items ...
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Creating a binomial die with constraints

I had an idea for making a custom 20-sided die that would have a binomial distribution of numbers printed on it, but I'm not sure exactly how to pick the numbers to put on the die itself. So we have a ...
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Statistical test for null hypothesis $\|p-q\|\le \epsilon$ for $k$-dimensional categorical data

Is there any known (asymptotic) statistical test for the null hypothesis $$\|p-q\|\le \epsilon$$ for $k$-dimensional categorical data independently taken from two societies for some given norm $\|\...
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How does the binomial coefficient relate to the combinations formula?

I have been having some problems reconciling the definition of "combinations" with the binomial coefficient. The definition goes something like: combinations compute the number of ways in ...
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Probability of m successes in n Bernoulli trials with unknown success probability Q∼Uniform(0,1)

I am working on a problem involving Bernoulli trials with an interesting twist. Here's the setup: $n$ independent Bernoulli trials are conducted. The success probability $Q$ is unknown and follows a ...
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4 votes
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Strong law of large numbers for $\mathrm{Bin}(n, p_n)$ variables

Massive edit to simplify the central question Suppose $X_n\sim \mathrm{Bin}(n, p_n)$ be a collection of independent random variables such that $np_n\to \infty$. Can we say that $Y_n:=X_n/np_n\to 1$ ...
Landon Carter's user avatar
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binomial distribution problemo

A fair coin is tossed n times. The histogram for the resulting binomial distribution is labelled 0, 1, 2, . . . , n on the horizontal axis, and each column is 1 unit wide. How many columns are ...
moon river's user avatar
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binomial question PROOF style [closed]

Take the number $n$ of stages of a binomial variable $X \sim B (n, p)$ to be fixed, and allow $p$ to vary. $p$ is not the random variable of this binomial experiment - $X$ is - but "allow $p$ to ...
moon river's user avatar
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Estimates of the sum of the biggest terms in a binomial expansion. [closed]

Perhaps this is an easy and standard result in the binomial distribution. Consider the expansion $(a+b)^n$ where $a=(\frac{1}{2} +\delta)$ and $b=(\frac{1}{2}-\delta)$ for some arbitrarily small $\...
Aritro Pathak's user avatar
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Is the distribution of sample mean of Bernoulli random variable a Binomial distribution?

As written in the title, is the distribution of sample mean of Bernoulli random variable a Binomial distribution? And I was taught that we can approximate Binomial distribution to normal distribution ...
비선형편미분방정식's user avatar
4 votes
4 answers
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Binomial distribution where successes increases the number of trials

I'm trying to calculate the probability of $k$ successes when the number of trials ($n$) increases by $6$ each time a success occurs (starting at $20$): $$ P(X=k) = \ ? \qquad\text{where}\qquad p=.05, ...
Salt's user avatar
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Cumulative distribution function of the Binomial Distribution for fixed probability of success is decreasing in the number of trials

Fix $0 \le p \le 1$. The cumulative distribution function of the Binomial Distribution $B(n, p)$, which counts the number of successes out of $n$ independent trials, each of which has probability $p$ ...
Colin Tan's user avatar
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Rate of decay of the Binomial cdf

Let $X \sim \text{Bin}(n,p)$ and $t \geq 0$. I believe the following inequality holds but I don't know how to prove it: $$ \text{Pr}[X \geq 2t] \leq \left ( \text{Pr}[X \geq t] \right )^2. $$ I have ...
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Calculating probabilities of binomial distributions for two methods

I'm struggling with this question on binomial distributions: A factory is considering two methods, I and II, of checking the quality of production of the batches of items it produces. Method I ...
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PGF of a Poisson-binomial

Let $N$ being a Poisson-binomial random integer with success rates $r_1,\dots,r_K \in [0,1]$. Since $N$ can be written as the sum of $K$ independent Bernoulli integers $N_i$ (each one with success ...
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Finding maximum value using expected profit with Binomially distributed demand

Let $D$ be Binomially distributed with mean $25$ and variance $20$. This means that $D\sim\text{Bin}(125,\frac{1}{5})$. I need to determine the optimal order quantity $Q$ which maximizes profit. For ...
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Probability of average of N samples of an unfair dice bigger than 5.

Imagine we have a dice: Side probability 1 0.1 2 0.1 3 0.1 4 0.1 5 0.2 6 0.4 We will throw the dice N times (ie. 7 times) And we want to calculate the probability that the average of the 7 ...
Oscar Flores's user avatar
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How to Determine if a Claim Should be Rejected in a Binomial Distribution

Context I ran into this problem in the book Fundamentals of Probability with Stochastic Processes (problem 5.1.7): "A manufacturer of nails claims that only 3% of its nails are defective. A ...
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Conditional Binomial Variables with co-related 'p'

$ X_1 \sim Bin(n , p1) , X_2 \sim Bin(n , p2)$ are dependent R.V. and, p1 and p2 are such that: $$ V \sim Uniform(a , b): $$ $$ p1 = P[V < r_1] $$ $$ p2 = P[V < r_2] $$ (1) Comment on the ...
Ricky Doo's user avatar
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Distributing persons into cars and autos [from Black Book]

I encountered a very interesting problem from Black Book. 14 persons depart from a place in 2 cars of capacity 4 each and 2 auto - rickshaws of capaity 3 each. Find the number of ways in which they ...
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Number of lottery participation needed to win a certain amount

Here is how a lottery works: There are 30'000 participants. Out of all of them, 1 wins 100\$, 10 win 10\$, 100 win 1\$, and 1'000 win 0.1\$. For each run of the lottery, a participant can only win ...
Vincent Tournier's user avatar
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Does the Poisson limit theorem talk about random variables or distributions?

I am confused about the way we take a limit below when saying a Poisson is a limit of binomial, and also whether we are talking about random variables in the limit, or distributions. The textbook says ...
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For $Y \sim$Poisson$(\lambda)$, is there necessarily an $X\sim$Binomial$(n, \frac \lambda n)$ defined on $Y$'s sample space?

I have learnt that the probability distribution of a binomial random variable $X \sim$Binomial$(n, \frac \lambda n)$ converges to the Poisson distribution with parameter $\lambda$ as $n$ goes to ...
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$95\%$ confidence for repeated bernoulli trials

For a known success probability $p$ of an event $X_p$, I know that the expected number of Bernoulli trials for the outcome to occur at least once is $1/p$.1 For example, the expected number of die ...
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standard error of an estimate of fraction of objects with a known binary probability

Lets have a group of N people that can be either men M or women W (or in general objects that can be 0 or 1). For each person $i$ we have an estimate of probability $P_i$ from <0,1> that it is ...
paskal's user avatar
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Sum and difference of independent binomial random variables

Suppose $X_1$ and $X_2$ are i.i.d. Binomial random variable where $X_1, X_2 \sim \mathrm{Bin}(n, 1/2)$. What can we say about the random variables $X_1 + X_2$ and $X_1 - X_2$? I know they are not ...
Xin Yuan Li's user avatar
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1 answer
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normal approximation to the binomial distribution in fair die

Suppose a fair die is independently tossed 1000 times. (a) Use the normal approximation to approximate the probability that the number 1 appears at least 180 times. (b) Use the normal approximation ...
edgar's user avatar
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How to find the generating function of a compound random variable?

I have the following compound binomial random variable: $B_n \sim \operatorname{Binomial}(X_n, 1-p)$, where $X_n$ is itself another random variable. This means that $(B_n \mid X_n = x_n) \sim\...
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How to calculate conditional probability involving the sum of a Poisson and a Binomial random variable?

I have the following process: $\{X_0, X_1, X_2, \ldots\}$ Here, $X_0 = 0$. $X_{n+1} = P_n + B_n$ where $B_n \sim \operatorname{Binomial}(X_n, 1-p)$ and $P_n \sim \operatorname{Poisson}(1)$ independent ...
Shatarupa18's user avatar
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Should I have discrepancies in the mean of a discrete distribution when estimating it's PDF using Inversion sampling?

I've been trying to create the Inversion sampling method in R for the distribution $Y \sim X_1 + X_2$, where $X_1 \sim Bin(100, 0.3)$ and $X_2 \sim Bin(100, 0.7)$. I haven't used ...
Rowan Harley's user avatar
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Determining value of expected Value formula that has extra variable inside.

i'm doing my homework and I got stuck at the question below and i'm assuming its under random variables because that is what i'm learning at the moment. for $0 \le p \le 1$ and $n = 2,3,4,....$ ...
Catt's user avatar
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Rolling a die, binomial distribution

I didn't get the exact thing that this question wants. Is the concept about the binomial distribution? I am not sure if I should calculate the probability or expected value. Also, this question means ...
Parande's user avatar
1 vote
2 answers
60 views

Probability differently coloured balls coincide in at least one box

I am trying to solve the following question: Suppose $k$ red balls and $l$ black balls are placed uniformly at random in $n$ boxes, where $n$ is much larger than $k$ and $l$. What is the probability ...
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Prove an inequality concerning binomial distribution

I hope to show that for a binomial distribution $X\sim Bin(n,1/2)$, $\mathbb{E}\log(X+1) \ge \log(n/2)$. This reduces to an inequality: for any $n\ge 2$, $$ \sum_{i=0}^n \log(1+i) \binom{n}{i}\left(\...
Will Cai's user avatar
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Which of the answer is correct? Combination vs Binomial Distribution

A class of 400 students having 220 women and 180 men. The class is divided into study groups of 10 students each. What is the probability that a certain group will have exactly 4 men? I have used the ...
Pratikkumar Bulani's user avatar
1 vote
1 answer
40 views

False Acceptance Probability in a binary hypothesis testing

There are two hypotheses about the probability of heads for a given coin: $\theta=0.5$ (hypothesis $H_0$) and $\theta=0.6$ (hypothesis $H_1$). Let $X$ be the number of heads obtained in $n$ tosses, ...
Raja Ali Riaz's user avatar
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Proof that if $Z = X + Y$, where $X\text{~Bin}(n, p)$ and $Y\text{~Bin}(m, p)$ and are independent, then $Z\text{~Bin}(n + m, p)$.

I had to prove this theorem. If $X\text{~Bin}(n, p)$ and $Y\text{~Bin}(m, p)$ are independent random variables then $$Z=(X+Y)\text{~Bin}(n + m, p)$$ My proof: If we consider $Z = X + Y$, then $E(Z) =...
A. Srivastava's user avatar
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Prove $\sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} {n \choose k} p^k (1-p)^{n-k}$

Prove that the following two summations are equal for any positive integers $i\leq n$, and any real number $p$ between $0$ and $1$: $$ \sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} ...
Yuzhen Feng's user avatar
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How to know the probability of the number of failures units in a cluster with single unit failure probability

I have a situation at work, where we need to release a product. We found an issue that occurs randomly. After a thorough analysis, we were able to root cause of the issue. The fix for the issue is not ...
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