Questions tagged [multinomial-distribution]

Questions in probability which includes more than one random variable

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entropy of weighted sampling without replacement

Let us have $k$ elements of weights $w_1,…,w_k$. We sample elements with weights without replacement. At the first, the probability of element $i$ to be sampled is $w_i / \sum_{l=1}^k w_l$. Then, the ...
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Parameter estimates and uncertainties associated with a Multinomial distribution

I have a process/model the depends on a single parameter $\lambda$ that generates $n+1$ outcomes. From $N$ events I can estimate the probabilities $\hat{p}_k$ from $N_k/N$ using a MLE for a ...
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On asymptotics of certain sums of multinomial coefficients

Given positive integers $n$ and $k$, set $$ S_{n,k}=\sum_{\substack{a_1+a_2+\dots+a_k=2n\\ a_i \in 2\mathbb{N},\,i=1,\ldots,k}}\frac{(2n)!}{a_1!a_2!\dots a_k!}, $$ where $2\mathbb{N}=\{0,2,4,\ldots\}$....
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Power average of multinomial distribution

Given a multinomial distribution $$ X = \left(X_{1},\ldots,X_{K}\right),\ \sum_{k = 1}^KX_{k} = n,\quad\mbox{and}\quad p_{1} = \cdots = p_{K} = \frac{1}{K}, $$ I wonder whether there is some ...
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Necessary condition for constrained optimization

Suppose $X=(X_1, \cdots,X_k)$ follows the multinomial distribution with a known size $n$ and an unknown probability vector $(p_1,\cdots,p_k)$. Find the necessary conditions for the solution to the ...
Nothing's user avatar
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Multinomial example from online game [closed]

Suppose that a multinomial distribution has 25 outcomes, the first 24 have chance $\frac{1}{465}$ and the final has $\frac{441}{465}$ chance. Find $n$ the number of trials for this such that the ...
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Multinomial distribution: probability that at least one outcome didn't occur

I'm trying to find the probability, that in a group of $N$ people, there are no people from at least one district with populations $n_{i}$ (for $i \in \mathbb{N}$ ranging from $0$ to $k$, where $k+1$ ...
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Finding probability from Dirichlet distributions under constraints

Suppose that I have the following: Three Dirichlet distributions of dimension k: distr1, distr2, distr3 A vector A of dimension 3 and a vector B of dimension k. Does anyone know about an algorithm ...
crixus's user avatar
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Multinomial probability calculation [closed]

$10$ cars choose uniformly at random between three parking lots ($A$, $B$, and $C$). Calculate the probability that none of the parking lots are empty and that there are exactly $5$ cars in lot $A$. ...
rolf's user avatar
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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 write the distribution of $(x_1,x_2)$ given $X_3=x_3$

Let $x=(x_1,x_2)$ be a trinomial distribution with parameters $(n;p_1,p_2)$. Deterine the conditional distribution of $(x_1,x_2)$ given $X_3=x_3$. Remember $X_3=n-x_1-x_2$. How do I write the ...
Brian Omondi's user avatar
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Whether $-p(\mathbf{y}|\mathbf{\alpha})$ is convex w.r.t. $\mathbf{\alpha}$ when the prior is Dirichlet and the likelihood is Multinomial?

Assume the prior is given as $$ \operatorname{Dir}(\boldsymbol{\mu} \mid \boldsymbol{\alpha})=\frac{\Gamma\left(\sum_{k=1}^K \alpha_k\right)}{\Gamma\left(\alpha_1\right) \cdots \Gamma\left(\alpha_K\...
Jonas Lionel's user avatar
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Sampling procedure to obtain groups with same categorical distribution

Imagine that we have $n$ (mutually-exclusive) sets of different sizes, $X_1, X_2, ..., X_n$. The total number of elements is $N = |X_1| + ... + |X_n|$. I want to partition these $N$ elements into ...
pterojacktyl's user avatar
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Sheldon Ross (Multinomial Distribution Problem 1.5)

Suppose that $n$ independent trials - each of which results in either outcome $1, 2, \dots, r$ with respective probabilities $p_1, p_2, ..., p_r$ such that $\sum_i p_i = 1$. Let $N_i$ denote the ...
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Marginal Posterior Distribution for Multinomial/Dirichlet Variables

Suppose that some data $(y_{1},\ldots,y_{J})$ are distributed multinomially with parameters $(\theta_{1},\ldots,\theta_{J})$ and that $\theta = (\theta_{1},\ldots,\theta_{J})$ has Dirichlet prior ...
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Accounting for an uncertainty in the number of categories of the multinomial distribution

Assume that we have an unfair die, and our task is to determine the probabilities of rolling a 1, a 2, a 3 and so on by rolling the die. Unfortunately we have no way of knowing how many sides the die ...
uLoop's user avatar
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Multiple dependent binomial random variables is a multinomial random variable? How to reconcile this with multinomial's requirement for independence?

The Wikipedia page for the multinomial distribution says the following: For $n$ independent trials each of which leads to a success for exactly one of $k$ categories, with each category having a ...
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N balls in a jar where the probability of the i_{th} ball being chosen is P_{i}. M students choosing one ball each with replacement. Probability?

So this question was given to my class by our CS Professor as a challenge. It goes as follows: There is a jar which contains $N$ balls and there are $M$ students. The probability of the $i$th ball ...
Prabhat Sharma's user avatar
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How to compute the expected standard deviation of a multinomial distribution?

Given is a multinomial distribution with $k$ mutually exclusive events with probabilities $p_1, .., p_k$. We draw a sample of size $n$ and get sample sizes $s_1, .., s_k$. The expected value for each ...
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Expected number of non-uniform draws until collision?

Edit May 9 -- high-level summary of the issue here. $R$ gives a good proxy for estimating collision time, with a slight undercount. Random matrices and graphs give distributions with longer time until ...
Yaroslav Bulatov's user avatar
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How do I compute the expectation value of a product of of random variable, pulled from separate distributions, but those distributions are correlated?

I want to compute the following expectation value:$$\langle \phi_i ~m_j \rangle$$ Where $\phi_i \sim \text{multinomial}(s_1,(p_1,...,p_N))$ and $m_j\sim\text{multinomial}(s_2,(\alpha_1,...,\alpha_N))$ ...
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Multinomial Distribution - Compute the probability of the sample containing 6 with grade

Consider a class with 100 students enrolled. Suppose that 30 achieved a mark over 70%,60 achieved between 50-69% and 10 achieved 0−49% . Let's take a randomly selected sample of 12 of these students ...
JMAbbott's user avatar
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1 answer
125 views

Multinomial distribution - every outcome at least m times

I'm drawing $k$ times independently and uniformly random from the the set $\{1,...,n\}$ (aka a multinomial distribution where all $p_i$ are equal). Is there a formula for (a good lower bound to) the ...
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Probability that at least $m$ molecules are in a given box

Imagine we have $N$ molecules that are distributed randomly into $K$ boxes. What is the probability that a given box contains at least $m$ molecules? (For example, we could have $N=10K$, $m=20$ and we ...
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Sum of the variance of individual element of multinomial distribution

Suppose I have a multinomial distribution with 10 outcomes and each has prob of 0.1 (i.e. weight of 10%). After each random draw, I can calculate the weight of the sample w1, w2... w10, and I want to ...
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Probability of a point in a square

Suppose I have on the map a square of dimensions 100 x 100 meters of which I know its center (by means of its latitude and longitude, (L,l)). Let's imagine that I am given a point of coordinates (X,Y) ...
Mel Schlichting's user avatar
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118 views

Show that the random vector follows a multinomial distribution and find it's parameters.

I am trying to show the following in the below setup, I have written my answers and approach below. I am having a hard time understanding the second and last part, especially the last part. Consider ...
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Example for writing the multinomial distribution as sum of multinoulli distributions

I am trying to understand the answer provided by @MichaelHardy in this thread (https://math.stackexchange.com/a/204094/554130). He explains how a multinomial variable can be viewed as sum of ...
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Multinomial Distribution - Special Case

Suppose you roll 6 ’fair’ dice. Let event A be the case where exactly two dice show a number less than 3 (1 or 2). Let event B be the case where where exactly two dice show a number greater than 4 (5 ...
Daniel Lader's user avatar
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Can I use a multinomial distribution to solve this problem?

I'm trying to write a simulation that models population dynamics a set of organisms occupying a specific niche. Let's say I have 3 different species each with populations $n_1, n_2, n_3$. Together ...
A-P's user avatar
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Variance and co-variance of bivariate multinomial distribution

Suppose we have two discrete variable $(X,Y)$, where $X$ takes values $x_i, i=1,2,...,k$ and $Y$ takes values $y_j,j=1,2,...,l$. Suppose we have total $n$ samples of $(X,Y)$, let random variable $Z_{...
Mingzhou Liu's user avatar
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Expected number of weighted dice rolls to achieve a certain outcome

Lets assume we have a weighted dice roll with sides $ \lbrace 1,2,3,4,5,6 \rbrace$ given $p(i)= p_i $ and $\sum_{i=1}^6 p_i=1$ for some $p_i \in \left(0,1\right)$. Denote by $X_i^{(t)}$ the random ...
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How to find the conditional expectation $\mathbb{E}[A|b]$ of a multivariate normal pair $(A,B)\sim \mathcal{N}$?

Suppose that $A,B$ are random Euclidean vectors and $(A,B)$ follows a multivariate normal distribution $$\mathcal{N}\left(\begin{bmatrix}E\\ F\end{bmatrix}, \begin{bmatrix}P & Q\\ R & S\end{...
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Multinomial distribution question

Let there be 12 different events with different probabilities of happening, each of which is known to us. How do we then find the probability that all the events happen at least once for a given ...
totlay's user avatar
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1 answer
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Multinomial distribution - closed form for $P(X_1 = X_2, \dotsc, X_{2n-1} = X_{2n})$

Let $n, k \in \mathbb N $ and $(X_1, \dotsc, X_{2n}) \sim \operatorname{Multinom}(2k, p_1, p_1, \dotsc, p_n, p_n)$. Is there a closed form of $$ P(X_1 = X_2, \dotsc, X_{2n-1} = X_{2n}) = \sum_{k_1 +...
Urh's user avatar
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Distribution of $X_1+ X_2$ when $X_1, X_2, X_3$ follows a multinomial distribution

The setting is that $X_1,X_2,X_3 \sim Multinomial(n;p_1,p_2,p_1+p_2)$. So the constraint is that $X_1+X_2+X_3=n$, and $p_1+p_2+(p_1+p_2) = 1$. I don't really understand how I should go about finding ...
gws's user avatar
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Cumulative Multinomial Distribution does not seem to add up to 1.00 for parameters >3

For a multinomial distribution $P = ( n! / (\prod_{i=1}^k {n_i!}) ) * ( \prod_{i=1}^k {{p_i}^{n_i}})$ If you sum the resulting multinomial distribution for every possible unique frequency for k=2 for ...
Anon's user avatar
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1 answer
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Want a hint for this multinomial distribution problem

Below is my attempt, but I failed to continue. The 2 dices are not necessarily identical but they are independent. Let $A, B$ denote the outcome of 2 dices rolling, then the pmf of $A, B$ would be $P(...
Cooper's user avatar
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1 answer
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Expected value of sample entropy of a dice

A dice has $K$ sides, each side has probability $p_k$ of coming up in a roll, such that $p_k \geq 0$ and $\sum_{k=1}^K p_k = 1$. Let $X$ denote the random variable corresponding to such a probability ...
Aleksejs Fomins's user avatar
1 vote
1 answer
398 views

Random point on hypersphere surface at a uniform distance of another point on the surface

Suppose I have an n-sphere of radius 1 centered in $(0,0,...,0)$, where each point on the surface represents a multinomial distribution: Given coordinates of a point $S=(x_1, x_2,...,x_n)$ on the ...
johacks's user avatar
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3 votes
1 answer
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Can we prove a trinomial distribution approaches a Gaussian

Consider a random walk with trinomial distribution where $i^{th}$ step is a random variable $X_i$ takes values $-\frac{1}{i}, 0, \frac{1}{i}$ with probability $0.3, 0.4, 0.3$ respectively. (A simpler ...
Shree's user avatar
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1 answer
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Calculation on a multinomial question

I'm working through the following multinomial question: Thirty items are inspected in a graded quality control procedure. The possible grades that can be awarded to an item are fail, major flaw, minor ...
Stackcans's user avatar
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1 answer
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Calculate the ML estimate in multinomial distribution

It is the process of calculating the maximum likelihood estimate {$\pi_j$} in a multinomial distribution. The multinomial log-likelihood function is $l(\pi)=\sum_j{n_j}{log\pi_j}$ $\partial l(\pi) \...
user913386's user avatar
13 votes
2 answers
278 views

Tied chess matches and the monotonicity of $\sum_{k=0}^n \binom{2n}{k,k,2n-2k} (pq)^k (1-p-q)^{2n-2k}$

In the upcoming World Chess Championship 14 games in the classical time format will be played compared to 12 in the previous matches. This change appears to have been made mainly to reduce the number ...
ComplexYetTrivial's user avatar
2 votes
0 answers
506 views

Fisher Information for multinomial distribution

Genotype AA, Aa, and aa occur with probabilities [$\theta^2, 2\theta(1-\theta),(1-\theta)^2$]. A multinomial sample of size n has frequencies ($n_1, n_2, n_3$). I try to derive a Fisher information. $...
user913386's user avatar
13 votes
3 answers
960 views

Guessing number of colors of beads in an urn

Motivation from cocktail bar Every time when I order the cocktail “Latex and Prejudice” (“Латекс и предубеждение”) in the Tesla bar in Saint Petersburg (Russia) the barkeeper selects by random a small ...
granular_bastard's user avatar
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1 answer
3k views

mean and variance formula derivation for multinomial distribution

I need a derivation of mean and variance formula for multinomial distribution. I tried to prove the formula, but I don't know what is meaning of expected value and variance in multinomial distribution....
joshua's user avatar
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1 vote
1 answer
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Conditional expectation of a component in a multinomial distribution: understanding of complex calculations

This is in the context of the expected value of a multinomial distribution in statistics, but I don't think that needs to be known for this specific question. I'm confused about how an answer in my ...
adam dhalla's user avatar
1 vote
1 answer
78 views

Zero count in a multinomial experiment

I have the following problem: n balls are dropped, independently, in n different urns and we want to calculate $\mathbb{E}V_{n}$, where $V_{n}$ is the number of urns with zero balls. I know that the ...
Mads C's user avatar
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0 answers
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MLE of multinomial distribution from multiple realizations

I observe multiple histograms as data points, each denoted as $n_{i}$. The total number of points is $N$, i.e., $i=1:N$. Each $n_i$ is $V$ dimensional, and $L_i = \sum_{v=1}^V n_{iv}$. $L_i$ can be ...
maktukmak's user avatar