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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/Binomial_distribution)

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Binomial distribution with nonlinear function of successes

Is there a closed form expression for the following expression: $$\sum_{j=1}^{N-1} {N-1\choose j} q^j (1-q)^{N-1-j} \frac{c-jd}{e+jd}$$ where $c$, $e$, and $d$ are some real numbers? I wonder if the ...
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Binomial Random Variable Intuition

I have the following question from a stats course about binomial distribution: A multiple choice test has 10 questions, each with 5 possible answers, only one of which is correct. A student who did ...
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the moment generating function of the negative binomial distribution

According to the textbook I use, it states that: $X$~$Neg.bin(x;k,\theta) = {n-1 \choose k-1}\theta^k(1-\theta)^{n-k}$ Which I have no problem. The problem arises when I try to find the moment ...
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Consistency of maximum likelihood estimator for negative binomial distribution

Let $X = (X_1,\ldots, X_n)$, where $X_1,\ldots,X_n$ are independent and have the same distribution: $$P_{\theta}(X_i = k) = \binom{k + r - 1}{k} \cdot \theta^r \cdot (1 - \theta)^k~,$$ with $k \in \...
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factorial moment generating function

I'm trying to get the factorial moment-generating function of a binomial random variable. I know that $F_X(t) = E[t^x] = \Sigma_xt^xp(x)$ so I get $\Sigma_xt^x{n \choose x}\theta^x(1-\theta)^1-x$ ...
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Sum of dependent Binomial distributions

In one of my classes, we stated that if $X_i$ are independent Bernoulli random variables with p proportion of success, then the distribution of the sum $\sum X_i$ is Binomial(n,p). I already proved ...
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Determine the error-probability of biased coin tosses using chernoff-bounds

Let's assume we have a biased coin with probabilities $\frac{4}{5}$ and $\frac{1}{5}$ and we don't know to which event (head or tail) the probabilities belong to. But we want to decide it by majority ...
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Finding $E[X^2]$ for $X \sim Bin(25,0.61)$

I got something rather new and I just wanted to make sure my way of thinking in this field is fine. Suppose $$X\sim Bin(25,0.61)$$ and we are asked to find: $E[X^2]$. So basically I treat this ...
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Likelihood interval for binomial counts

I have an assignment question regarding a "likelihood interval" that I don't really understand. The question asks to consider counts of $X_i$, with $i\in \{1,...,N\}$, modelled as independent ...
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Probability using binomial distribution

I am trying to le-learn probability theory and I am solving the following problem: A probability that a manufactured device has $3\%$ or more deffects is $p=0.02$. If a company buys $5$ devices, ...
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Rounding or decimals in binomial distribution

if in a problem you are to find the probability of passing a course of more than x-percent people and the number is decimal, do you round to the nearest one or for every new natural number started you ...
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Beta-binomial distribution for scaled and translated Beta

Recall, that a binomial distribution in which the probability of success at each trial is randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can calculate ...
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Probability and Statistical Modelling Proof (Negative Binomial into Poisson)

The Negative Binomial RV $X$ models the number of trials until the $r$-th success in a sequence of independent Bernoulli Trials with probability of success $p$ in each trial. So, if $q = 1 - p$, $$P(...
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Probability Problem: Combination of Poisson and Binomial

A hitchhiking gentleman waits by the roadside for cars to pass by that will take him to his destination. The daily amount of cars that pass by him that are going to his destination behaves in the ...
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How to simplify this sum

I'm having a lot of issues with this question. Ive done problems where I need to find the coefficient or the constant using the binomial theorem but I'm not sure how to even begin doing this. I looked ...
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Binomial distribution and hypothesis test.

Sign test: A new experimental drug is given to patients suffering from severe migraine headaches. Patients report their pain experience on a scale of 1 – 10 before and after the drug. The ...
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Simplifying summation of binomials

I am working on a proof and think that the following equality holds but am unable to prove it: $$ \sum_{k_1=0}^{m} \sum_{k_2=0}^{m-k_1} \dots \sum_{k_{d^2/2}=m-(k_1+k_2+\dots+k_{d^2/2-1})}^{m-(k_1+k_2+...
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Unfair coin question

Supposed that you flip an unfair coin with the probability of heads being $p({\rm heads}) = 3/4$ and the probability of tails being $1/4$ a total of ten times. How do you find the possibility of ...
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Sum of parts of binomial distributions

I'm going to draw two 0/1-coded random sequences from 1 binomial distribution $Bin(n=20,p)$. Depending on the sum $\Sigma$ over the elements of that sequences I classify the sequences as either $A$ or ...
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How will p-values behave when fitting normal/Poisson to binomial?

I know p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(Poisson). Now suppose I take random binomial samplings and fir normal(Poisson) to it, for ...
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How to solve this factorial equation?

How to solve this factorial equation? $$\frac{n!}{(n-6)!} = 350.418$$ or to give you the original equation: $$0.146 = \binom{n}{6} \times 0.45^6 \times 0.55^{n-6}$$ Sorry, but I've no idea about ...
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Binomial equality [closed]

The advisor asks to verify that the coefficient of $$x ^ n$$ in the development of: $$(1+x)^{2n}+x(1+x)^{2n-1}+x^2(1+x)^{2n-2}+......+x^n(1+x)^n$$ is equal to $$\binom{2n+1}{n}$$ I tried for ...
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Show that estimators are consistent and show which is more efficient

I have two independent binomial $X_1 =$ binomial $(n,p)$ and $X_2 = $ binomial $(2n,p)$ we assume that $n$ is known but $p$ is an unknown parameter I want to show that $P_1= 1/3n(X_1+X_2)$ and $...
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Coin flipping limit

Let $S_{n}\sim\text{Bin}\left(n,p\left(n\right)\right)$ where $p\left(n\right)$ is the unique solution to the equation $\delta\left(p\left(n\right),n\right)=0$ with $\delta$ being continuous and ...
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Binomial Probability Number of Trials Formula and Code

This is an interview question I've had. Your number space is between 0 to N. You can only draw M random samples from 0 to N (N >= M). How many times (T) would you need to draw from your number space ...
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Finding the pmf of a r.v. S ~ Poisson's multinomial distribution. [duplicate]

Notation # categories $= c$. # trials $= t$. Side $i$ = $si$. Random vector $= S = \left[S_1\;S_2\;\ldots\;S_c\right]^T$. # occurrences of $si$ vector $= s = \left[s1\;s2\;\ldots\;sc\right]^T$. ...
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Finding pmf for 4 variations of an urn problem.

I would like to test my understanding of distributions by verifying my answers to 4 deliberately related questions. Urn R has 5 R marbles. Urn G has 11 G marbles. Urn B has 4 B marbles. Random vector ...
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Finding the probability that each apple in a randomly selected bag has a mass less than 105 grams.

I am practising normal distribution exam type questions but I am stuck at this one: The masses of individual apples sold in a food store are normally distributed. The supplier who provides the ...
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prove that $Y_1 \sim Bin(n_1,\pi)$ and $Y_2 \sim Bin(n_2,c\pi)$ is an exponential family

A statistical model for a data set y is an exponential family , with canonical parameter vector $\theta= (\theta_1,\theta_2,..\theta_k)$ and canonical statistic $t(y) = (t_1(y),t_2(y),..t_k(y))$ if $...
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Binomial distribution and Number of Successes

I was wondering if there is a formula relating the number of successes and the sample size in a binomial distribution. Is there such a formula?
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Binomial Distribution of Random Variable

The mean of defective blades supplied in packets of $10$ is $1$. in how many packets of this make out of $1000$ packages would you expect to find at least $4$ non defective blades. Answer given in my ...
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stochastic domination of binomial distributions

Consider the process of adding $k$ balls into a box: each ball $b_i$ will be added with its own probability $P(b_i \in B)$ for the first $k' = \lfloor \rho k \rfloor $ balls we know that $P(b_i \in B)...
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Asymptotics of the minimum of Binomial random variables

Let $Y=\min\{X_1, X_2 \cdots X_k\}$ be the minimum of $k$ iid Binomial $(n,1/2)$ random variables. I'm interested in the asymptotics of $Y$ (distribution, or mean and variance) for large $n$ and $k = ...
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How to find a confidence interval of a binomial distribution using a simulated random sample?

I have a random sample of 1000 values of deviates from binomial distribution with n = 52 and p^ So I have 1000 values from the distribution. How can I find a 95% confidence interval for the true ...
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True accuracy of neural network

My goal is to calculate the probability to correctly classify an object if I make $k$ predictions on slightly different images of it. The predicted class would then be the one that was predicted the ...
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Series limit involving Binomial coefficients

Consider the parameters $v_{a}$, $v_{b}$ be such that $0<v_{a}\leq v_{b}$ and $c>0$. I have an equation involving the Binomial distribution that I need to solve with respect to $p(n)$: $\sum_{k=...
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Binomial distribution confidence interval using a random sample?

I am working with the binomial distribution Bin(52, 0.82) I am looking to find the confidence interval for the observed test statistic of 44 successes. I have generated a random sample of length ...
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Getting a probability $> 1$ in hypothesis test

I have the following hypothesis question. A soft drinks company claims that of all consumers buying their product, $82 \%$ prefer the light version of the drink. To test their claim, data were ...
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Combinatorics floor summation $\sum_{i=0}^{\lfloor n/2\rfloor}\binom n{2i}p^{2i}(1-p)^{n-2i}=\frac12((2p-1)^n+1)$

While solving a problem I came across a rather interesting identity, and I do not see how I could prove it. $$\sum_{i=0}^{\lfloor n/2\rfloor}\binom n{2i}p^{2i}(1-p)^{n-2i}=\frac12((2p-1)^n+1)$$ Any ...
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The efficacy of mumps vaccine is about 80% that is 80% of those recieving the mumps vaccine will not contract the disease when exposed.

The efficacy of mumps vaccine is about 80% that is 80% of those recieving the mumps vaccine will not contract the disease when exposed. Assume that each person's response to the mumps is independent ...
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Find the least upper bound of a binomial distribution

In this problem we are given a random variable $X$ of binomial with $n=119 ,p=0.5$. I have calculated $\mathbf{E}[X]=59.5$ and $\mathbf{Var}[X]=29.75$. Looking for $D$ in order to have $\Pr[|X-\mathbf{...
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Varshamov-Gilbert, question about proof with probabilistic method

Let $d \in \mathbb{N}$ such that $d \equiv 0 \pmod{4}$, and take $S_1, \dots, S_d \sim Bernoulli(0, 1)$. Define for $v \in \{0,1\}^d$ the ball $B(v, d/2) = \{w \in \{0,1\}^d : \|v - w\|_1 \leq d/2\}$....
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The expected value within a time interval

Let's say that I have a group of 71 people and I'm trying to find the expected value of how many of their birthdays fall within a specific 6 day time interval (June 12-17). Would this type of ...
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Probability of drawing a total of at least x amount of black balls from different boxes?

Suppose we have two boxes containing a 100 balls. Box A has 45 black, 55 white balls and Box B has 30 black and 70 white balls. There is no trick with the boxes and the drawing process is completely ...
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Coin flipping and Bayes' theorem… but where does binomial theorem come in?

Consider the following question: You are face with two identical coins. One is fair, and the other comes up Heads 90% of the time. You flip coins, which results in THHHTHHHTH (seven heads, ...
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$\sum_{k=0}^{n} {n \choose k}^2 (1+x)^k (1-x)^{n-k}$ as a function of $(1-x^2)$

I am trying to show that $$\sum_{k=0}^{n} {n \choose k}^2 (1+x)^k (1-x)^{n-k}$$ can be expressed explicitly as a function of $(1-x^2)$. I am wondering where to start with this, as anything I have ...
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Solve the following problem by using the Binomial formula

If $n=10$ and $p=0.60$ find $P(X\geq 3)$. The formula we've been given: $P(X=x) = \binom{n}{x} \cdot p^x \cdot q^{n-x}$, where $n$ is the number of trials, $p$ is the probability of success and $q ...
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The value of houses in a certain district is normally distributed with mean $\$235420$ and standard deviation $\$28724$ (see below)

The value of houses in a certain district is normally distributed with mean $\$235,420$ and standard deviation $\$28,724$. a) Find the percentage of houses in this district with values from $\$200,...
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What is O in binomial distribution entropy?

I have searched everywhere but can't find an answer. What is "O" refers to in this equation? $${\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)$$
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limit of sum of binomials

I want to find the limit of the following equations: $\lim_{n\to +\infty} \sum_{k=0}^{n-2} \alpha_{\!_{(k)}} {{n-1}\choose{k}} [1-p_{\!_{(n)}}]^{k} [p_{\!_{(n)}}]^{n-1-k} \quad (1)$ where the ...