# 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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### 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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### 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 $p$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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