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Questions tagged [multinomial-distribution]

Questions in probability which includes more than one random variable

The multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts of each side for rolling a k-sided die n times. For n independent trials, each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

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Show that the multinomial distribution has covariances ${\rm Cov}(X_i,X_j)=-r p_i p_j$

If $(X_1,\cdots, X_n)$ is a vector with multinomial distribution, proof that $\text{Cov}(X_i,X_j)=-rp_ip_j$, $i\neq j$ where $r$ is the number of trials of the experiment, $p_i$ is the probability of success for the variable…
User 2014
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43 cookies are randomly given to 10 children. What's the probability each child receives at least 2 cookies?

I wanted to ask 1) if I've solved this puzzle problem correctly, and 2) if there is a shorter or more elegant approach. There are 43 cookies to be given out at random to 10 children. What is the probability that each child gets at least 2…
ctesta01
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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 of draws by allowing the players to take more…
ComplexYetTrivial
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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 interesting photo$^1$ and attaches it with a…
granular_bastard
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Balls into bins: expected number of bins attaining the maximum

We are given a multinomial distribution with $k$ bins and $n$ balls. The number of balls is at most the number of bins, i.e., $\sqrt{k} \le n \le k$. The probabilities of throwing a ball into a speficic bin are monotone non-increasing, i.e. $p_1 \ge…
CuriousGuy
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Proof concerning the multinomial distribution

Despite a long search I was not able to find a rigorous proof of the fact that a random vector having a multinomial distribution with parameters p (the vector of probabilities) and n (the number of trials) can be written as the sum of n independent…
Marco
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Is the probability that the sum of random integers is divisible by $k$ equal to $1/k$?

There are $n$ cards which have numbers $1$~$n$ on each. You pick $m$ cards from it, and you don't put it back once you pick from it. Is the probability that their sum is divisible by $k$ always $\dfrac 1 k$, while $k|n$? If not, how do we…
RDK
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What is the continuous distribution version of multinomial distribution?

I am trying to model a distribution, on the number of occurrences of an event in a 24 hour time span. Right now, I discretize the 24 hour time span into hourly intervals, and each hour is taken as a categorical outcome, and I count the number of…
Michael
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Entropy of the multinomial distribution

What is the entropy of the multinomial distribution? To fix notation, let us define $n > 0$ as the number of trials, $p_1, \ldots, p_k$ as the probabilities of each of the $k$ possible outcomes and $X_1, \ldots, X_n$ as the outcomes. Recall that the…
PianoEntropy
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Multinomial Distribution: Maximum Possibility

I am reading Albert N Shiryaev's Probability. There is one question from Chapter I §2. Problem 2: Show that for the multinomial distribution $\{P(A_{n1},\ldots, A_{nr})\}$ the maximum probability is attained at a point $(k_1, \ldots, k_r)$ that…
ygao
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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 $2\mathbb{N}=\{0,2,4,\ldots\}$. According to the answers of Special sum of…
S.Z.
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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 collision than what you'd expect by looking at…
Yaroslav Bulatov
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Generalization of Coupon Collector's Problem

Suppose I have a bag containing three different marbles: red, green, and blue. I am drawing a single marble from the bag each time with replacement. I would like to know how many times, on average, do I need to draw marbles from the bag until I have…
user752164
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Distributing balls into bins randomly

Problem: If $n$ balls are distributed at random into $r$ boxes (where $r \geq 3$), what is the probability that box $1$ at exactly $j$ balls for $0 \leq j \leq n$ and box $2$ contains exactly $k$ balls for $0 \leq k \leq n$ ? Answer: Let…
Bob
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Upper bound for probability of no collision for two balls-into-bins processes

I am considering the following probabilistic balls-into-bins model. There are $n$ bins and two types of balls. For each type, there are $\rho$ balls. Each ball independently lands in bin $i$ with probability $p_i$, where $\sum_{i=1}^{n} p_i =…
Idra
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