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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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Binomial distribution calculate probability of pikes - is my solution correct?

In a lake, there are two types of fish: trouts and pikes. Let p = 0.7 be the proportion of trouts in the lake. We pick 20 fish at random with replacement. Let X be the number of trouts. a) What ...
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Given X ~ N(0,1).Find P(X>0) and P(X<0)

All I understand here is that it is a question of binomial deviation where the particular case is of standard deviation i.e ( I familiarise it with the bell curve.an image has been attached.) .I am ...
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Root of a plynomial in (0,1)

Define $$f_K(x)=\sum_{i=K+1}^{2K} \binom{2K}{i}x^{i-1}(1-x)^{2K-i}.$$ How to show that $qf_K(x)-f_K(1-x)$ has exactly one real root in $(0,1)$ for any $q > 0$ and $K \geq 1$. The proof for $q=1$ ...
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not able to understand a probability concept

Suppose there are patients suffering from a particular lung disease. Either lung is diseased with a probability of 0.1. How to find the probability of exactly n lungs being diseased ? n = { 0,1,2 } ...
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Analogue to the beta-binomial distribution for sampling without replacement?

The beta-binomial distribution characterizes the number of successes in $n$ trials, but where the probability of success at each trial is unknown or random. However, suppose that you had finite ...
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Max of conditional Negative Binomial

Suppose $X|K = w$ is a Negative Binomial with parameters $r$ and $q$. K follows a Binomial Distribution with parameters $m$ and $p$. I want to calculate the expected value of $$Z = max(X_1, X_2,...,...
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Conditional Moment Generating Function of a Negative Binomial

Suppose X conditioned on K follows a Negative Binomial Distribution i.e. $X|K = k_i \sim NB(r-k_i, q)$, where $r$ is a constant and $K \sim Bin(m, q)$. I'm trying to calculate the MGF of X. So far, I ...
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Expected number of Failures within K trials for Binomial RV

Question: Given a binomial random variable with probability of success p and probability of failure (1-p). What is the expected number of failures given that a success was observed within k trials ...
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Proof of a Binomial expression summation

Let $x,y$ be probabilities and $n$ is some integer. Show that: $\displaystyle \sum_{n_0=1}^n \sum_{m=0}^{min(n_0-1,n-n_0-1)} \binom{n_0-1}{m}\binom{n-n_0-1}{m}x^m(1-x)^{n_0-m-1}y^m(1-y)^{n-n_0-m-1} ...
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Probability Of Machine Working

A complex machine is able to work if at least 3 of it’s 5 components work. If each motor independently functions for a random amount of time with density given by $f(x) = \frac{x}{e^x} , x>0$, ...
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A proving question based on binomial theorem [closed]

$$C_0-C1(a-1)(b-1)(c-1)_+C_2(a-2)(b-2)(c-2)+.... (-1)^nC_n(a-n)(b-n)(c-n) $$=0 I tried to solve this problem by using multinomial theorem but was not able to proceed further please help me out.
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Showing the sum of binomial independent variables follows a binomial distribution using moment generating functions

So I'm trying to solve the following problem: Show that if $X_i$ follows a binomial distribution with $n_i$ trials, and probability of $p_i=p$ for $i = 1,2,3...n$, and the $X_i$ are independent, then ...
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Binominal distribution OR

I got this exercise: In a multiple choice quiz there are 5 questions and 4 choices for each question (a, b, c, d). Robin has not studied for the quiz at all, and decides to randomly guess the ...
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Binominal distribution “fewer than”

I have this exercise: We learned in Exercise 3.26 that about 90% of American adults had chickenpox before adulthood. We now consider a random sample of 120 American adults. (a) What is the ...
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Determine whether the following $H_0$ can be accepted or not using hypothesis testing

$H_0 : p=0.5$ $H_1 : p > 0.5 $ where p is the probability of heads from a coin flip. Let $W_1$ be the number of heads from 10 coin flips and and $W_2$ be the number of heads from 1000 coin ...
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Finding probability using binomial distribution.

I am given a question that there is a quiz consisting of 20 true or false questions. Probability of answering correctly is 0.95. What is the probability that a given student answers all questions ...
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Why does the use of Jeffrey distributions does not satisfy the likelihood principle?

It's commonly used as an example, the Bernoulli experiments seen as a binomial and negative binomial random sample, with a posterior distributions $$\Pi _J ^1 (\theta) \propto\theta^ {-\frac{1}{2} }(1-...
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Probability with replacement, question on balls in urn.

An urn contains $n$ balls numbered 1 through $n$. If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously, find $P\{X=k\}$, $k = 1,.....m$, where $X$ is the ...
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Probability of getting an even number of sixes in $n$ throws of a die

A fair die is thrown $n$ times. Show that the probability of getting an even number of sixes is $\frac{1}{2}[ 1 + (\frac{2}{3})^{n}]$, where $0$ is counted as even number. My solution. I have ...
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Given that $X \sim \operatorname{Binomial}(n,p)$, Find $\mathbb{E}[X(X-1)(X-2)(X-3)]$

Given that $X \sim \operatorname{Binomial}(n,p)$, Find $\mathbb{E}[X(X-1)(X-2)(X-3)]$. It is suggested that I can transform it into \begin{align} \mathbb{E}[X(X-1)(X-2)(X-3)] &=\sum_{k=0}^n k(k-...
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is there a formula to “invert” the binomial distribution - for simulation purposes

My apologies if this should be in one of the programming sites rather than the mathematics one... I decided it was theoretical enough to post here. Feel free to move if someone with authority ...
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Help With Statistics and Distributions!!

So the question asks: Airlines find that each passenger who reserves a seat fails to turn up with probability 0.01 independently of other passengers. Consequently, Bryanair always sell 100 tickets for ...
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How to prove this binomial distribution

Assume that X1 ∼ P(µ1) and X2 ∼ P(µ2) are independent. Prove that the conditional distribution of X1 under the condition X1 + X2 = N to be the binomial distribution B(N, p),where p = µ1/(µ1 + µ2). How ...
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Probability of outcomes that do not match binomial distribution

Let's say I have an urn with 100 balls marked with numbers from 1 to 100. I would like to figure out how often I draw at least 2 balls that marked with [1-5] after 5 successive drawings(with ...
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1answer
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Price of European Call Option

A ECC called $X$. The buyer of $X$ has, at any time $t$, the option to receive a European Call option, $C$ or to receive an European Put option, $P$. Both with the same maturity $T$ and strike price $...
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calculating confidence intervals for a weighted binomial distribution

We have n binomial distributions {$b_i$} - each with m trials, and a probability of success $p_i$ somewhere in the range [0,1]. Also, each binomial distribution $b_i$ is assigned some weight $w_i$, ...
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What is the expected result for the number of heads obtained in this coin/dice flipping example?

Suppose you roll one fair six-sided die and then flip as many coins as the number showing on the die. (For example, if the die shows 4, then you flip four coins.) Let Y be the number of heads obtained....
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Binomial Distribution Application in determining conditional probability

Teams A and B play a series of games, with the first winning 3 games being declared the winner. Suppose that A independently wins each game with a probability 'p'. Find the conditional probability ...
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Variance of binomial distribution with binomial distribution of $n$

Accoring to Wikipedia, the variance of a binomial distribution $B(n,p)$ is given by: $Var(B(n,p)) = np(1-p)$ Now what is the variance of a bionomial distribution $B(n,p)$ where $n$ itself is defined ...
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Type-I vs. type-II error in statistical hypotheses testing

Let us consider standard statistical hypotheses testing: $$\alpha=P\{\text{type}-I \text{ error}\}=P\{\text{Rejecting } H_0 \text{ when }H_0\text{ is true}\}$$ and $$\beta=P\{\text{type}-II \...
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calculating variance and expectation of unknown binomial variables over a window

I have $2^m$ independent random variables. All have binomial distributions, each with $m$ samples. The probability of success for each binomial distribution is somewhere in the range $[0,p]$ (...
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What's the probability that B will catch 3 fish before A catches 3 fish

This is a basic question, but I do not completely understand. A and B are both catching fish at times of independent poisson processes with rates $1$ and $2$ respectively. What is the probability B ...
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A (possibly biased) coin with heads probability p

[This non-trivial undergraduate level statistics question is self-answered and adds to the knowledge pool.] Question A (possibly biased) coin with heads probability $p$ is tossed $n$ times, and the ...
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Let X be a binomially distributed random variable with mean 2 and variance 4/3. Tabulate the probability distribution of X.

I'm practicing for a test that I'm writing tomorrow and one of the past questions was: Let X be a binomially distributed random variable with mean 2 and variance 4/3. Tabulate the probability ...
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Hypergeometric Distribution & Probability of selecting without replacement & n Percentage

I will ask my question through an example. There is 100 man in N. 20 man "have card", 80 man "does not have card". If i randomly pick 50 man, what is the percentage of "man with cards" in this 50 ...
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Binomial distribution confusion

Whats the difference between pbinom(2074000, 4247000, 0.5) and pbinom(2074, 4247, 0.5), why do they give differing values when the proportions are same?
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Prove that given distribution is a binomial distribution

I'm taking an elementary probability course and came across this problem. I know what a binomial distribution is and its properties. But I have no idea how to prove if a given distribution is one. ...
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Probability that more than 220 heads appear in 400 coin flips

If I toss 400 fair coins, estimate (to the nearest whole percentage point) the probability that more than 220 heads appear. Here is my thought process: This is a binomial variable. We know that $p = ...
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Binomial distribution with random variable parameter

Im stucked with this exercise: Let $p\in(0,1)$, $T\in\mathbb{N}$ and $d,u\in \mathbb{R}_+$ such that $d<u$. Further let $(\Omega,\mathcal{F})=(\{u,d\}^T,\mathcal{P}(\Omega))$ be a ...
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1answer
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Two components with different probabilities of failure.

I am stuck with an exercise. It says we have a system consisting of two different kinds of components A and B with probabilites to fail of 10% and 20%, respectively. To build the system we use ...
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2answers
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There are 50 misprints in a book which has 250 pages, find the probability that page 100 has no misprints? (Use theoretically correct distribution)

My question is where this should be modelled as a binomial distribution problem or a Poisson distribution problem. Any hint/advice helps, thanks in advance!
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Binomial Random variable to prove to expression

Consider a binomial random variable $X\sim\operatorname{Binom}(n,\theta)$. Show that $f_X(x+1)=f_X(x)\left(\frac{n-x}{x+1}×\frac{\theta}{1-\theta}\right)$. Show that $f_X(x+1)>f_X(x)$ iff ...
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Probability Question for Defective Bulbs (Verification)

A company manufactures bulbs. The probability that a randomly selected bulb, after it is shipped to a customer, is defective equals $0.01$ A) In a shipment of $100$ bulbs, what's the probability ...
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Can the binomial distribution be solved for k?

I am working on a problem where, given the binomial probability density function: $$X(p,n,k) = \left(\frac{n!}{k!(n-k)!}\right)p^k(1-p)^\left(n-k\right)$$ I need a function $Y(p,n,x) = k$ where $p, n$ ...
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Multiplying Binomial N and P for Expectation?

The problem given is X ~ Bin(5,.5) and the E(X) was calculated by just multiplying 5 and .5 together to get 2.5. I thought to calculate the expectation we would take values X=0 (p 0) + X=1 (p 1) etc. ...
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1answer
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Calculation of random variable probabilities

I have a question about how to solve this exercise: The monthly sales of a certain consumables store are distributed evenly, with average 4000e and standard deviation 1200e, and the estimated expenses ...
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1answer
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How to calculate a random variate of a binomial distribution for very large n?

Say I want to generate a random number $x\in [0, 128^{64}]$ with the expected value being 64. The trivial answer seems to be to roll a random vector $v = [0,1)^{128^{64}}$ with the answer then being $...
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Probability relating the sum of tilted distributions

Say I have $iid$ Bernoulli random variables with probability $p$ $$P(X_i = 1) = p$$ I derive the tilted probability $X_i^{(\lambda)}$ such that: $$P(X_i^{(\lambda)} = 1) = \frac{e^{\lambda}p}{e^{\...
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Calculate average retransmissions in communications

In an network, when transmitting a packet, it can be rejected by a first router with a probability p. Then It can also be rejected by a second router with the same probability p. We have something ...
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Binomial Distribution: How to know the values of n, p and x from a worded problem?

The Problem: A tour operator organises a trip to Las Vegas. The tour operator knows that the probability of people withdrawing is 0.08 (independent event) so accepts 22 bookings even though only 20 ...