Questions tagged [discrete-distributions]
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19
questions with no upvoted or accepted answers
2
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0answers
23 views
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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, ...
1
vote
1answer
27 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:...
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0answers
19 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 $...
0
votes
1answer
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 $P_X(x < x_0 \mid K, N)$?
0
votes
0answers
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 ...
0
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0answers
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 ...
0
votes
0answers
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 ...
0
votes
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}
...
0
votes
0answers
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 =...
0
votes
0answers
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)=(...
0
votes
0answers
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, ...
0
votes
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 ...
0
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0answers
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 ...
0
votes
0answers
32 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 ...
0
votes
0answers
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 ...
0
votes
1answer
68 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))...
