Questions tagged [multinomial-distribution]

Questions in probability which includes more then one Random variable

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Dirichlet-Multinomial Model Maximum Likelihood Estimation

Given a Dirichlet Prior $p_1...p_k$ ~ Dir($\alpha_1 ,..., \alpha_k)$ where $\alpha_1, ... ,\alpha_k = \alpha$ and given $X_1,..., X_n$ ~ Multinomial(n, $p_1,...,p_k$) what is the MLE estimator of ...
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Estimation of the alpha in a Multinomial-Dirichlet Model

I have a question regarding the Multinomial-Dirichlet model. Given a Dirichlet Prior, and a Multinomial likelihood. How can I estimate alpha (the parameter of the dirichlet) through a MLE in closed ...
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9 views

Does Chi Square Square statistic always works for multinomial hypothesis testing?

The $𝒳^2$ statistic has been frequently employed in the hypothesis testing of the multinomial distributions. But after looking at the derivation of such a process(where they use Taylor expansion and ...
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How to describe explicitly the multinomial random vector

Describe explicitly the Multinomial random vector $N: \Omega\mapsto\mathbb{R}^{m}$ Also, define $(\Omega,\mathscr{F})$ Notation: ($\mathscr{F}$) is the sigma-algebra. I think that "Describe ...
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15 views

How to find MLE for multinomial distribution and expectation of $X_1$ and $X_2$?

There are 3 types of flowers that can grow from planting a seed. $$P(\text{Daisy}) = \theta_1$$ $$P(\text{Rose}) = (1-\theta_1)\theta_2$$ $$P(\text{Sunflower}) = (1-\theta_1)(1-\theta_2)$$ the total ...
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1answer
18 views

Multinomial distribution pooling

Suppose i have a random vector $N_1,...,N_{2n}$ following a multinomial distribution with $k$ trials and an even number of (mutualy exclusive) outcomes $2n$ such that the probabilities of each outcome ...
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2answers
48 views

Joint distribution of random vectors on a multinomial distribution

I am having some problems with this proof, I don't know how to start, I think that it's related to the joint distribution. Honestly, I don't have a clue. I will appreciate any help Let $\textbf{N}$ ...
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76 views

Multinomial Distribution — How to calculate percentiles?

I've read the rules and searched but I do not even know what I'm looking for. Here is my problem: Suppose I have a bag containing three different marbles: red, green, and blue. I am drawing a single ...
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28 views

Is it possible to construct the joint distribution of multinomial distribution?

Is it possible to construct the joint probability density of multinomial distribution? May you assume two multinomial distributions are independent and multiply their distributions to obtain the ...
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30 views

Multinomial Maximum Likelihood Estimation

Let $(X_1, X_2, X_3, X_4)$ be a random vector from a multinomial $$\left[n,\frac{1-2\theta+\theta^2}{5},\frac{\theta(2-\theta)}{5},\frac{\theta(2-\theta)}{5},\frac{(1-\theta)^2}{5}\right]$$ find the ...
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15 views

How could I compute probabilities over sums of multinomial samples?

I am taking some samples from multinomially identically distributed random variables $X^i$ : $X^i = (X^i_1, ..., X^i_m) \sim Multinomial(n, p_1,...,p_m)$ Denote the vector: $X^* = \sum_i X^i$, i.e. ...
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44 views

Find $P[X-Y=1]$ given $X,Y \sim Bin(20, 1/5)$

You, your parents, your sister, go to visit grandma for her birthday. Grandma made a cake for the party. If she puts $20$ raisins in the cake at random in the cake, and she divides the cake into $5$ ...
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Choosing a discrete proposal distribution for a multinomial process in a particle filter.

I have a particle filter and I know the target distribution is modelled via the multinomial distribution. I had a very similar multinomial distribution for my proposal distribution. However, I ended ...
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31 views

$n$ blue letters and $n$ pink letters to place in $n$ envelopes

Given $n$ blue letters and $n$ pink letters (in total $2n$ letters) to place in $n$ envelopes. on each envelop there's an address written, and the letters also have $n$ different addresses. In each ...
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44 views

If there are $k$ places and we have $m$ different numbers repeated $k_1, k_2, \dots, k_m$ times, then the number of ordered samples is …

I am currently studying the textbook Statistical Inference by Casella and Berger. In a section on combinatorics, the authors state the following: In general, if there are $k$ places and we have $m$ ...
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81 views

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 $...
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34 views

normal Distribution of the Sum of independent Variables \ Using Z table

We throw Fake Cube(Containing $1,2,3,4,5,6$) $~300$ times . Probability of geting $6$ is $\frac{1}{2}$ Probability of geting $1,2,3,4,5~$ is $~0.1$ What is the probability that that the Sum : $~~S \...
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80 views

Binomial-Multinomial Distribution

In probability theory, what is the difference between binomial and multinomial distribution? According to my understanding, binomial distribution has 2 outcomes, whereas multinomial distribution has >...
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2answers
701 views

Multinomial distribution to Binomial distribution?

We have a multinomial distribution on $X=(X_1,\ldots,X_r)$ with the parameters $(n,p_1,\ldots,p_r)$ Now we will have to determine the distribution of $X=(X_1+X_2,X_3,\ldots,X_r)$ Does that mean we ...
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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 ...
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1answer
123 views

Is this a Multinomial distribution?

I find it hard to notice when do I have a Multinomial distribution and if its possible to "transform" problems into a Multinomial distribution problems. For example I have the following exercise: ...
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2answers
970 views

Average distance from origin in a random walk on the integer number line

In a random integer walk along a number line (each step 0.5 probability of moving right and 0.5 probability of moving left), what is the average distance from the origin during the walk? Other ...
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945 views

Is there a Continuous Multinomial Distribution??

In Multinomial Distribution, we have \begin{align} f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1\mbox{ and }\dots\mbox{ and }X_k = x_k) \\ \\ & {} = \begin{cases} { \displaystyle {...
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1answer
513 views

Definining a multinomial distribution/conditional probability function.

I'm having trouble finding an expression for the function: $$f_{X_1+X_2,X_2+X_3|X_2+X_4}(·,·|t)$$ The notation is a little confusing for me but I know that $(X_1,··· ,X_{10})$ follows a multinomial ...
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1answer
4k views

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 ...
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1answer
227 views

Simplified Multinomial Distribution

I am working with a simple case of the multinomial distribution, as follows: There are $k = 8$ different possible outcomes, each occurring with equal probability $p = \frac{1}{8}$. What is the ...
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2answers
1k views

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 ...
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1k views

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 ...