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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/...

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Estimator for binomial distribution

I have a question from my introduction to mathematical statistics book. I'm working on the following problem. We have an urn with a ratio of white balls to black balls of $\frac{p}{1-p}$. We count ...
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Normal and Binomial application in one problem (tricky)

Some people claim they have ESP therefore we perform an experiment on 15 such people. Four cards are kept and they need to select the blackened card. Each of these people are given 500 chances to ...
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A probability question related to Bernoulli distribution

I'm self-studying probability and I bumped into this question: suppose a physical trait (such as eye color) is based on one pair of genes and suppose that $d$ represents a dominant gene and $r$ a ...
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Given Y Bin ~ (10,0.4) and Z Geo ~ (0.3) find P(Y+Z=2)

Given Y Bin ~ (10,0.4) and Z Geo ~ (0.3) find P(Y+Z=2) Why is the solution to this P(Y=0)P(Z=2)+P(Y=1)P(Z=1)? Why doesn't P(Y=2)P(Z=0) need to be included in this solution?
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How to prove that maximum occurs at x=n/2 when n is even

I am stuck with this question: Using the recursion formula of a binomial distribution and for $\theta = \frac{1}{2}$: $b(x+1;n,\theta) = \frac{\theta(n-x)}{(x+1)(1-\theta)}*b(x;n,\theta)$ Show that ...
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Why is a portion of the expected value for a binomial random variable equal to 1? [on hold]

Trying to follow a proof for the expected value of a binomial random variable equal to np. I'm stumped on this step: why does $$\sum_{j=0}^m\binom{m}{j}p^j(1-p)^{m-j}$$ result in 1?
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Expected number of cutting edges of a graph

The graph with $E$ number of edges and $V$ number of vertices. Divide the vertices into two groups $A$ and $B$ with the independent probability to be $p$ and $1-p.$ If the connected vertices are in ...
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I don't see how the binomial theorem relates to the principle of inclusion and exclusion?

I'm learning discrete maths as a hobby at the moment and I got stuck when the tutor starting relating the binomial theorem to the principles of inclusion and exclusion. The video I was watching is ...
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Binomial Calculating Probability of Airline Tickets

Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 115 passengers. The probability that a passenger does not show up is 0.05, ...
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Likelihood function with inequality

Suppose $Y_1, \dots, Y_n$ are i.i.d. bernoulli random variables. Also, $Y=\sum Y_i \sim binom(n, \theta)$ and we have a prior beta distribution $\theta\sim beta(a,b)$. I want to compute $P(\theta>0....
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The number of ways in which $n$ distinct items can be divided among $r$ groups

The number of ways in which $n$ distinct items can be divided among $r$ groups such that no group contains less than $m$ and not more than $k$ items $(m<k)$ is Please solve this question.I am ...
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Convergence of ratio of binomial tail probabilities goes to 1

I'd like to claim that $$ \sup_{j\in H_0}\sup_{n\theta\leq c\leq n\theta+b\sqrt{n\log m}}\left\vert\frac{G_{0,j}(c)}{G_0(c)}-1\right\vert\to 0 $$ where $G_{0,j}(c)=P(X_j>c\mid\theta_j,j\in H_0)$...
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Show $\mathbb{P}(S_n \geq a+b) \leq \mathbb{P}(S_n \geq a) \mathbb{P}(S_n \geq b)$ for sum of independent Bernoulli random variables

Let $X_1,...,X_n$ be random variables independent and identically distributed, where $X_i \sim \textrm{Bernoulli}(p)$. Define $$\displaystyle S_n = \sum_{i=1}^n X_i$$ Show that $P(S_{n} \geq a+b) ...
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Fast binomial sampling

This question might appear stupid, but I haven't found how to do it: I want to simulate a sampling on a large number of samples (very large n, in the distribution $b(n, p)$, and count the results. ...
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Probability of at least two, three and four events happening out of 9 samples?

Help please!!! 40% of the population likes bananas. 60% of the population likes apples. Out of a sample size of 9 people, what is the probability that at least 2 like bananas? What about at least ...
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How can I get N trials from binomial distribution (Edited)

$$ \sum_{k=0}^{17}{_NC_k}\times 0.1^k\times 0.9^{N-k}<0.004 $$ How can I get a $N$ from above inequality?
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How can I get $N$ trials from binomial distribution?

$$ \sum_{k=1}^{15}{_NC_k}\times(0.1)^k\times (0.9)^{N-k} > 0.9996 $$ Is there a way to get $N$ from this? I was looking for a lots of materials but I couldn't find.
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Binomial pair of dice to get a total of 9

What is the probability of getting a $9$ exactly once in $3$ throws with a pair of dice? What I tried: There are five possibilities: $(4,5), (5,4), (3,3), (6,3), (3,6)$ Therefore, Probability is $5/...
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Determine probability [on hold]

If $20\%$ of the bolts produced by a machine are defective, determine the probability that out of $4$ bolts chosen at random a) $1$ b) $0$ c) Less than $2$ bolts will be defective.
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Question related to binomial distribution?

This may seem like an odd question but I appreciate any input. I am playing a game where you are trying to promote a player's skill level. You get tokens to attempt the promotion and you can attempt ...
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probability of a 50/50 event occurring 15/20 times divided by the probability of a 75/25 event occurring 15/20 times

The material I am working with: http://personal.vu.nl/a.f.de.vos/primer/primer.pdf Article describes probability of a 50/50 event occurring 15/20 times divided by the probability of a 75/25 event ...
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Expectation value : trials with and without replacement

Two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is Direct Method $$ P(X=0)=P(\text{no ace})=\frac{{}^{48}C_2}{{}^{52}C_2}=\frac{48*...
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Simulation with product of proportions in a biological process

I am modeling a biological process that uses plant tissue as the starting material to generate plants. The number of plants (k) successfully made with the inputs (n) is a small proportion (p). The ...
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Mind Boggled : Not exactly a fixed number of Trials, but Independent Events?

And so, I have a math question for my stats homework that goes like this : A car dealer has a list of 15 cars. The probability of selling one car during a typical week is 40%. The chance of selling ...
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Probability of Error going unnoticed in a noisy channel

I had a doubt regarding a selected question in the book Introduction to Probability by Joseph K. Blitzstein and Jessica Hwang. I seem to have missed out a key point in my approach to the problem. ...
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MGF for Binomial Distribution

I’m learning towards an exam I have. In one of the questions I've being asked to compute the MGF for the binomial distribution. My answer is slightly different from the official answer published by ...
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Calculating probability that soccer player scores in a certain game with binomial distribution

Since Ronaldo just came to play in Italy but didn't manage to score on his first 2 matches, I wanted to give an estimate of the probability that he scores tonight. In order to achieve this (in a ...
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How does randomly creating a subset (from a set) affect the probability (that was originally applicable for the item in the set) of the items in it?

I have the following question from a book. A lot has 10% defective items. Ten items are chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is? And ...
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What is the expected value and standard deviation of a weighted binomial distribution?

I'm working on a genetics problem and am using binary representation of the genes i.e. they can be 0 or 1. With 20 genes, I have an expected additive genetic value (i.e. mean) of 10 with standard ...
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Conditional probability and binomial distribution

We have a system of 10 satellites. The probability that a satellite works on a maintenance day is 0.85 and the probability that it works on a non-maintenance day is 0.95. Maintenance days are 30% of ...
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Confidence intervals for proportions - why isn't the Bessel correction used in estimating the standard deviation?

When calculating confidence intervals for a population with standard deviation σ unknown, σ is estimated using the sample standard deviation S, which uses the Bessel correction to more closely ...
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Why do we use ${n\choose k}$ for a binomial distribution instead of ${n+k-1\choose k}$?

I am trying to get my head around this. In my understanding a binomial distribution uses replacement and ${n\choose k}$ precisely states that there's no repetition and that's not the case with a coin ...
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Showing Independent Cases in Binomial Distribution [closed]

Can someone please give me a hint on how to solve part 2 in the following question? I am not sure how to even begin... A professional proofreader has a $98 \%$ chance of detecting an error in a ...
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Calculating Cumulative Binomial Probabilities

I was given the question to find $P(10\leq X\leq 12)$ with $n=15$ and $p=0.666$. Does this mean that $P(10\leq X\leq 12)=P(X\leq 12)-P(X\leq 10)$? If so, I got the answer $0.9206-0.5959=0.3247$ yet ...
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I don't understand something about Laplace Theorem (binomial distribution)

Here is what I have for it: Laplace's theorem: Let $S_n$ denote the number of "successes" in $n$ Bernoulli's trials, and let $p$ be the probability of success $p\in(0,1)$. Then $\forall a,b\in \...
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Showing that estimator is minimax

I have the following question. Let $X\sim \text{Bin}(n,p)$ and consider estimating $p\in(0,1)$ with loss function, $$ L(p,\hat{p})=\left(1-\frac{\hat{p}}{p}\right)^2. $$ I need to show that the ...
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hypothesis testing - probability of type I and type II error - power of the alternative

Manufacturer of pharmaceutical products has to decide about the recovery from a certain disease for a new medication on basis of samples. For the test $H_0 : θ_0 ≥ 0.90$ versus $H_1 : θ_0 < 0:90$, ...
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Asymptotic behavior of combinations: approximating Hypergeometric by Binomial

A form of the hypergeometric distribution is $$P(X=x)=\frac{\binom{Np}{x}\binom{Nq}{n-x}}{\binom{N}{n}}$$ where $N\equiv$ total number of elements of the sample space $p\equiv$ probability of ...
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What bound does Chebychev’s inequality give for $P(Y ≥ 3 \text{ or } Y ≤ 1)$?

Suppose you flip four fair coins. Let Y be the number of heads obtained. (a) What bound does Chebychev’s inequality give for $P(Y ≥ 3 \text{ or } Y ≤ 1)$? $E(Y) = \sum_{y=1}^{4}yP(Y = y) = \sum_{y=1}^...
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Find a bound to the mean of a sum of Bernoulli variables

Let $\delta_i$ be a set of independent Bernoulli variables with $\mathbb{E}(\delta_i)=p$ for all $i=1,\ldots,n$. Let $$X = \sum_{i=1}^n |1-\frac{1}{p}\delta_i|$$. I get that $$\mathbb{E}(X)=\sum_{k=0}...
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Complicated probability question - Can you figure out the composition of the marbles in this urn?

There is an urn filled with millions of fancy marbles. Anytime a marble is drawn from the urn, an identical version is immediately put back into the urn by the manufacturing technology. The marbles ...
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Binomial distribution approaches to Poisson or Normal?

I'm reading two books and they say differently. In a binomial distribution $X \sim \text{Bin}(n,p)$, if $n \to +\infty$, $X$ approaches to Poisson distribution $\text{Po}(np)$. The other book says $...
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How to solve the probability of the binomial distribution sequence below?

Let $x_0$, $x_1$...$x_n$ be a sequence of independent random variables. $x_i = 1$ has probability $p$ and $x_i = 0$ has probability $1-p$. Let $k$ be the smallest integer such that $x_k = x_{k+1}$. ...
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Type 1 error condition in one tailed statistical hypothesis test

Consider the following classical statistical test setup: One assumes a coin to be unfair in the sense that heads, say, occurs more frequently than tails. Thus we set $H_0: p\leq\frac12$ as null ...
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What is what between binomial many flip outcomes and statistical observation?

I am trying to get my head around normal distribution which evolves as a good approximation for a binomial problem (like coin flips). Theoretical outcome: When I have say 50 flips, there is a ...
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Bernoulli vs. Geometric distribution

Suppose that in a sequence of independent bernoulli trails, the number of failures up to the first success is counted. FIND: What is the frequency function for this random variable? attempted ...
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The best way to make and update a Multinomial distribution?

I want to fit a distribution to a set of data I have. My questions are: How can I know the best distribution that can be fitted. The expectation is that the final distribution be a Binomial or a ...
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How is salary a binomial distribution?

In an online course here, the author presents a problem that salary is normally distributed, provides mean and variance. As per my understanding, when no of Bernoulli trials are sufficiently large ...
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Statistics: “hybrid” of Poisson and Binomial distribution

A salesperson has a 20% probability of selling a product during each call. The average rate of calls made per hour is 20 and follows a Poisson distribution. a) Find the probability that there are 3 ...
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Derivation of univariate normal distribution and how it approximates binomial distribution problems

Can some one kindly explain the derivation of normal distribution (univariate) and how it could be approximated for binomial distribution (coin flip problem)? I have searched for hours but could not ...