Questions tagged [multinomial-distribution]
Questions in probability which includes more than one random variable
118 questions
3 votes
1 answer
120 views
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 ...
- 210
-1 votes
1 answer
157 views
Expected value of a product of elements of a multinomial
From this question, i get that, given a random vector $ (X_1,...,X_m) \sim M(n;p_1,...,p_m)$, the expected value of the product of two variables is $E[X_1X_2] = n(n-1)p_1p_2$. It is possible to obtain ...
- 3
0 votes
0 answers
112 views
Variance of multinomial distribution
Context: I am trying to find $\text{Corr}(X,Y)$ where $X$ is a random variable with given $\text{Var}(X)=v$ and $\mathbb E(X)=e$. And $Y$ is a random variable with multinomial distribution $Y\sim\text{...
- 3
12 votes
2 answers
306 views
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 ...
- 499
6 votes
4 answers
333 views
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 $\...
- 3,433
1 vote
1 answer
75 views
Generalizing Bernoulli to Binomial without summing
A Binomial random variable is a generalization of i.i.d. Bernoulli random variables by summing them, arrived at by performing discrete convolution on their distributions. A Multinomial random variable ...
- 197
0 votes
1 answer
75 views
Geometric distribution with multiple failure types
I stumbled upon a variation of the Geometric distribution in which a failure may be of one of $n$ types and I wonder if this distribution has a name, what its properties are and what it has been ...
- 1
1 vote
0 answers
57 views
How to calculate standard error of a partial sum of balls in bins
As part of an experiment, I independently throw $M$ balls with equal probability into $N$ bins, resulting in $m_i$ balls in the $i$th bin. I then randomly choose $n$ of those $N$ bins and sum the ...
- 11
0 votes
0 answers
61 views
Expected number of draws from urn (with 3 sets of color balls) to get 2x as many red balls as blue balls.
I am computing the expected value of this event: suppose we're sampling with replacement from an urn, where the likelihood of red balls is $p$, the likelihood of blue balls is $q$, and the likelihood ...
- 1,528
0 votes
1 answer
72 views
Probability that of a plurality vote is correct with multiple options
There are many questions 1, 2, 3 as well as Condorcet's jury theorem that address finding the probability that a majority vote over a set of responses is correct when responses are generated by $k$ ...
- 302
0 votes
1 answer
76 views
Can you use multinomial distribution to analyze games like Stellaris?
The way Stellaris works is that its AI operates off a script: when the AI is evaluating what to do, it will choose its course of action randomly, with the probability of a certain action being ...
14 votes
3 answers
3k views
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 ...
- 514
4 votes
2 answers
304 views
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 $2\mathbb{N}=\{0,2,4,\ldots\}$....
- 610
1 vote
1 answer
96 views
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 ...
- 4,072
0 votes
1 answer
202 views
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 $k+1$ ...
- 120