Questions tagged [discrete-distributions]

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Maximum entropy discrete distributions with specified mean

Consider a discrete distribution on {1, ...,n}, with mean given as $m$, what is the maximum entropy distribution? I know it takes the form $p_{X}(k)=ar^{k}$ and is a geometric distribution when n is ...
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46 views

Measure how good is the discrete approximation of continuous random process?

If continuous random random process approximated with discrete version - how to measure how good is the approximation? One possible way is to compute and compare the moments. But for some probability ...
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15 views

How to prove or disprove that a complete sufficient statistic exists?

We have a discrete random variable which takes values with probabilities $p, q, p+q$ and $r$. I want to construct a complete sufficient statistic based on a single observation from this distribution, ...
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18 views

MLE and binary variables

Suppose $z$ and $y$ are discrete random variables taking values 0 or 1. The distribution of $z$ and $y$ is given by $$P\{z=1\}=\alpha$$ $$P\{y=1|x\}=\frac{e^{\gamma x}}{1+e^{\gamma x}}\\ z=0,1$$ Here $...
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53 views

What distribution is $P_X(x \mid K, N) = \frac{1}{ ( 1 + N)^K} {x + K -1 \choose x} \left( \frac{N}{1+N}\right)^x$?

What kind of distribution is the following: $$ P_X(x \mid K, N) = \frac{1}{ ( 1 + N)^K} {x + K -1 \choose x} \left( \frac{N}{1+N}\right)^x $$ and how can I find $P_X(x < x_0 \mid K, N)$?
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31 views

Test for null hypothesis: the rank of each variable is uniformly distributed

Assume the following situation: I have $N$ samples of $k$ continuos variables, $$x_n = (x^n_1,...,x^n_k).$$ I do not trust whether I can combine the continuos variables across samples, but I want to ...
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13 views

Distribution of Independent but Non-Identically Distributed Geometric Random Variables?

What distribution describes the sum of independent but non-identically distributed geometric random variables? That is, (abusing notation a little) if $\{X_n \sim Geometric(p_n)\}_{n=1}^N$, what ...
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35 views

joint PDF of continuous and discrete random variables

Given exponential a random distribution X with PDF $f_X(x)=\lambda e^{-\lambda x}$ and a random variable Z with the PMF $p_Z[z]=0.5, z= \pm1$, I am trying to find the PDF of $Y=ZX$ (I also know that Z ...
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1answer
48 views

Sufficient Statisitics and Discrete Distributions

I am trying to master minimal/complete sufficient statistics, however I am having trouble when the distributions are discrete and involve indicator functions. Here is my 3 part question: Let $X$ be a ...
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37 views

Generating correlated discrete random variables

Suppose that we have $q_t \in \{-1, 1\}$ where $\mathbb{P}(q_t = -1) = \mathbb{P}(q_t = 1) = \frac{1}{2}$. Further, assume that \begin{align} Cor\left( q_t, q_{t-k} \right) = \begin{cases} ...
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23 views

In poisson distribution why does setting r = mean_value not output 50%?

If I want to find probability of 10 cars passing highway checkpoint in 60 seconds where on average 10 cars pass in 60 seconds. Assuming there is no rush hour I can use Poisson distribution So lambda =...
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21 views

If $Y$ is continuous and $X$ is discrete, how to write the joint density of $(Y,X)$?

If $Y$ is continuous and $X$ is discrete with a finite number of points in the support, how to write the joint density of $(Y,X)$? For example, to write the joint density function evaluated at $(Y,X)=(...
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13 views

Are discrete maximum entropy distributions uniquely determined by their energy function?

I read this somewhere but can't find a reference, I'd love to be pointed in the right direction. I'm specifically interested in discrete Boltzmann distributions with interactions at different orders, ...
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1answer
53 views

guessing a number between 1 and 100

Person A chooses an integer between 1 and 100 at random, then B can guess that number in (at most) 7 attempts, i.e. $\log_2(100)+1=7$. What if now A chooses an integer from a distribution that is ...
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34 views

Maximum-likelihood histogram from noisy data

Given a sequence of noisy observations $\{x_k\in\mathbb{R}\}$ and a set of thresholds $\{t_i\in\mathbb{R}\}$ we can bin the observations using the thresholds to create a histogram. However, since we ...
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31 views

Approximating a distribution with an integer histogram

Given a distribution $f:[0,a)\rightarrow\mathbb{R}$, is there a simple algorithm by which to find a sequence $\{h_i\in\mathbb{N_0}\}$ such that $f(x)$ is approximated by $h_{floor(x)}$ as a histogram ...
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35 views

Factorial moment bound for discrete Binomial distribution

I need to compute the upped bound for the tail (survivor) probability $P(X \ge t)$ for the discrete Binomial random variable $X$. I could use Chernoff bounds, however according to this paper [1] the ...
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1answer
67 views

Bayes test function with a discrete prior

Let X be exponential with mean θ. Consider testing H0 : θ = 1 versus H1 : θ = 2 with a single observation. Loss function: 0-1 Loss function. So the risk of the test function φ is R(1, φ) = E1 (φ(X))...
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19 views

What are the limmitations of reparametrization gradients for discrete random variables? (Gumbel-softmax)

We know that one approach for re-parametrizing gradients for variational inference is taking the Gumbel-softmax estimator proposed in [1] and [2]. In [3], that is a talk on SVI, D. Blei at around 29:...