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Estimating Smooth Density Field from Limited Sampled Data

I want to estimate a “density field”, specifically $P(y|x, m)$, for binary labels $y$ associated with 2D points characterized by spatial coordinates $m$ and additional spatio-temporal features $x$. ...
Xaume's user avatar
  • 71
0 votes
0 answers
60 views

What is this hybrid(mixed) random variable’s variance?

X ∼ Uniform(a,b), a<b (Discrete) where f(x)=1/n where n=b-a+1 and Y ∼ Uniform(c,d), c<d (Continuous) where g(y)=1/d-c. X and Y are independent. Let z = x - y. I was able to find the E(Z), ...
raffaello.sanzio's user avatar
1 vote
1 answer
96 views

Calculation of multivariate probability mass function

How to calculate the following multivariate probability mass function: $P(X_1-X = n, X_2-X = n, ..., X_{N-1}-X = n)$ Where $n$ and $N$ are positive integers, and $X_i$ and $X$ are iid random variables ...
Francesco's user avatar
0 votes
0 answers
30 views

Probability of 2 discrete samples coming from same distribution

I would like to know how to calculate the probability that 2 discrete samples come from the same distribution, and if so, which one is the distribution they are coming from. Let's say we have 3 ...
Oscar Flores's user avatar
2 votes
1 answer
52 views

Existence of Distribution with Given Multivariate Marginals

Consider discrete random variables $X_1,\cdots, X_n$, and let $D$ be their joint distribution. For each subset $S\subseteq[n]$ let $D_S$ be the marginal distribution $(X_i)_{i\in S}$. Fix $k<n$. ...
AAA's user avatar
  • 121
1 vote
0 answers
54 views

Exact Likelihood ratio statistic for discrete distribution

Suppose that the random variables in a sample $Y_1, Y_2, \ldots, Y_n$ are iid with values in $[0,1]$, and that an investigator knows that the underlying probability density $f_Y(y)$ has the form $f_Y(...
Stats_Rock's user avatar
0 votes
0 answers
24 views

Understanding the Difference between Bernoulli Distribution and Binomial Distribution

In my recent study, I conducted 67 measurements and recorded 11 successful outcomes. Now, I am seeking clarification on the appropriate formulas to calculate the measurement error. Should I use the ...
DmitriBolt's user avatar
2 votes
2 answers
67 views

How to model virus spread over distinct days?

Say I want to model a simple virus spreading over 45 people with a transmission probability of 5%. If I choose my population to be exclusively hospital workers, assuming they work every day, I could ...
barker's user avatar
  • 261
1 vote
0 answers
14 views

discrete distributions [closed]

Please help with explanations. I tried with Pr(U=U) = 1-Pr(U<u) and similarly for V but i am getting the wrong answer. The entity X is a discrete random variable with the following distribution X: ...
Nisha Sheshashayee's user avatar
0 votes
0 answers
27 views

Generating samples from a categorical proposal distribution with specific output (sample) structure

I'll caveat this question with that I do not think what I want to do is possible but suggestions in the right direction would be most helpful. Now my data consists of vectors of length six, where each ...
Astrid's user avatar
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1 answer
304 views

Conditional expectation of a continuous random variable given a discrete random variable

For a problem I am working on I need to compute the conditional expectation of a continuous random variable $T$ given a discrete random variable $K$. I already derived a formula for the joint ...
stats19's user avatar
  • 61
1 vote
1 answer
61 views

Models for a discrete numerical response with only 5 distinct values

I have a response variable that takes on only 5 distinct values: -15, -5, 0, 5, 15. These values are obtained in a clearly categorical manner, meaning: if event A happens, then the outcome is 5, if ...
UsDAnDreS's user avatar
  • 139
0 votes
0 answers
40 views

Hypothesis testing for two samples from discrete distributions

There exists a classical method for solving a certain computational problem related to random sampling. It is the "gold standard", so to speak. I'm working on an algorithm that aims to solve ...
swineone's user avatar
3 votes
1 answer
86 views

What kind of transition probability matrix indicates dependence/independence?

Suppose we have two discrete random variables $X$ and $Y$, both of which take values from $\{1,2,...,k\}$. $Y$ is generated from $X$ via a transition probability matrix (also known as the stochastic ...
Mingzhou Liu's user avatar
2 votes
1 answer
38 views

Can we preserve dependence in discretization?

Suppose we have two categorical random variables $X$ and $Y$, with both $k$ levels. We assume $Y$ is generated causally from $X$ ($X \to Y$). We also assume causal Markovian is satisfied, which means $...
Mingzhou Liu's user avatar
5 votes
2 answers
137 views

Quantifying the confidence that the most sampled outcome is the most probable outcome

Let us say we have a cheater's six-sided die, which we can assume to be unfair with an unknown probability distribution. We want to know the most likely roll with this die, and so we roll it $N \gg 6$ ...
Codename 47's user avatar
2 votes
1 answer
112 views

Is it possible to generate a Pareto distribution with dice?

So I know that there's a really easy way to generate a normal distribution with dice (simply add them). Is there a way to generate a Pareto distribution?
user379945's user avatar
2 votes
1 answer
77 views

conditional probability related to drawing coins with 2 properties

I'm not quite familiar with conditional probability, and is having difficulty coming with a solution of the following problem: There are 3 coins in a box, and each of them is associated with 1 of the ...
user21's user avatar
  • 231
5 votes
1 answer
108 views

Discrete probability distribution involving curtailed Riemann zeta values

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Var}{\operatorname{Var}}$We define the discrete random variable $X$ as having the probability mass function $$f_{X}(k) = \Pr(X=k) = \zeta(k)-1, $...
Max Muller's user avatar
1 vote
0 answers
34 views

Is it possible to conduct a Chi-Square Test to determine the distribution of a binary uniform distribution?

Suppose a data with a large sample size has a binary categoric variable (i.e. True/False, etc) and it is assumed that the null hypothesis of the sample follows a discrete uniform distribution. Is it ...
Farrel K.B.'s user avatar
1 vote
0 answers
54 views

Bayes' rule for mixed discrete and multiple continuous random variables

Consider the Bernoulli random variable $Z$ that takes $z=1$ with probability $1/2$ and $z=0$ otherwise. A different random variable, $X$, is defined on $x \in [0,1]$ has the following conditional ...
Kenneth's user avatar
  • 11
0 votes
0 answers
218 views

Relationship between the distribution of weighted sum of a multinomial distribution and other well-known distributions

Following is the explanation of multinomial distribution from a source of lecture notes (by Yen-Chi Chen). The multinomial distribution is a common distribution for characterizing categorical ...
martian03's user avatar
  • 111
0 votes
1 answer
93 views

Methods for comparing discrete probability distributions?

What are some ways to see how different two discrete probability distributions are? I have two samples that can only take value from 1-4, in increments of 0.5, and I'm trying to compare their ...
user373876's user avatar
1 vote
0 answers
23 views

Criterion for quality of MCMC proposal in a simple discrete setting

Let's say I have a discrete distribution $p$ over n states that I want to sample from, but I only know a similar probability distribution $q$. However, I can obtain unnormalized probabilities $c \cdot ...
Jannis's user avatar
  • 310
0 votes
1 answer
31 views

How to measure the 'conservation' in a discrete probability distribution

I have a discrete probability distribution: [0.05, 0.05, 0.05, 0.25, 0.6] which represents the probability of a character(given by the index in the vector) to appear. How can I define a measure ...
GIONII's user avatar
  • 1
0 votes
0 answers
107 views

Compare 2 binned non-parametric histograms with hypothesis test

According to this guy here: https://towardsdatascience.com/how-to-compare-two-distributions-in-practice-8c676904a285 and the following papers he mentions inside his article, and more specifically the ...
nowhere's user avatar
  • 157
1 vote
0 answers
94 views

Any alternatives to Jackknife estimate for acceleration constant a in BCa bootstrap CI

I am trying to implement the bias-corrected and accelerated (BCa) bootstrap confidence interval (CI) for parameters from discrete distributions. The way to compute the acceleration constant "a&...
Lydia2kkx's user avatar
0 votes
0 answers
263 views

Find PMF when X and Y are two discrete Independent random variables

Let X and Y be two independent discrete random variables with PMFs $$\begin{equation} P_X(k)=P_Y(k)= \begin{cases}\frac{1}{5}, & \text{for}\ x= 1,2,3,4,5 \\\ 0, & \text{otherwise} \end{cases}...
Win_odd Dhamnekar's user avatar
6 votes
3 answers
589 views

Alternative formula for the Bernoulli pmf?

If I understand correctly, a Bernoulli pmf just needs to assign a probability $p$ if there is a success $(x=1)$, and $1-p$ otherwise $(x = 0)$. Rather than the usual formula, can't the following ...
Steve's user avatar
  • 661
0 votes
0 answers
6 views

A discrete distribution arising in a shorter sub-array problem [duplicate]

I have a list $L=\{x_1,x_2,\cdots,x_N \}$ of $N$ random integer numbers $x_i\in[a,b]$, $b\gt a$, $a,b\in\mathbb{N}$, $N\gg b$ and they follow a discrete uniform distribution. I need to scan the list ...
Alessandro Jacopson's user avatar
1 vote
1 answer
127 views

Proving independence of discrete variables and the product of them

Given that P(A) and P (B) are independent and \begin{equation} $P(A=1)=\frac{1}{2}$ $P(A=-1)=\frac{1}{2}$ $P(B=1)=\frac{1}{2}$ $P(B=-1)=\frac{1}{2}$ There is a random variable $C = A \cdot B$, C is ...
MrMercury's user avatar
1 vote
0 answers
74 views

Is this discrete distribution with Catalan numbers as coefficients named/characterized (or a special case of a named/characterized distribution)?

I have a complex stochastic model, and I want to (among other things) determine whether the output is over-dispersed, compared to what we would expect under a simpler model. Which means that it would ...
Chris Henry's user avatar
1 vote
2 answers
615 views

Upper bound of P[X < Y]

X and Y are independently distributed discrete random variables. is it possible to find an upper bound for P[X<Y] that is always less than or equals to 1?
BTM's user avatar
  • 175
0 votes
0 answers
4k views

We have a 4 sided die, a 6 sided die and a 12 sided die. We roll a die twice and get 1 and 5. What's the probability that we rolled the 6 sided die?

I'm trying to solve this question but I'm not sure about my thinking. So I think what I need to compute is : $$P[\text{rolled the 6 sided die} | \text{got numbers 1 and 5}]$$ I tried to do it with the ...
student-t's user avatar
2 votes
0 answers
160 views

Confidence intervals for integer parameters

I'm interested, purely out of curiosity, in what methods can be used to calculate confidence intervals for discrete integer model parameters. As an example, consider the model (which I can flesh out ...
Eoin's user avatar
  • 9,137
8 votes
2 answers
786 views

Regression model for integer response

Let the response be $Y_i \in \mathbb{Z}$ and the covariate $X_i \in \mathbb{R}^p$. For counting data where $Y_i$ are restricted to be nonnegative, we have Poisson regression or negative binomial ...
waic's user avatar
  • 110
0 votes
0 answers
226 views

Ways to measure deviation from a discrete uniform distribution [duplicate]

I'm looking for a way to characterize the deviation from a discrete uniform distribution. Example: 50 balls are distributed over 10 urns. In the most equal case, all urns get 5 balls. In the most ...
cosine's user avatar
  • 1
1 vote
1 answer
48 views

Construction of statistics of a discrete distribution

I have the following problem: we consider an i.i.d sample $\mathbf{X} = (X_1,...,X_n)$ of the discrete set $\{1,...,N\}$. An agent has to infer the probability distribution of $X_i$. I wanted to use ...
MiKiDe's user avatar
  • 113
3 votes
1 answer
63 views

Which is the correct solution to the hypothesis testing: $H_0 : \lambda =65, H_1 : \lambda >65$ , $X$ is a Poisson ($\lambda$) ,$\alpha=0.05$

Given the following hypothesis test: $H_0 : \lambda =65, H_1 : \lambda >65$ , where $\lambda$ is the parameter of an $X$ distributed as a Poisson $\alpha=0.05$ . We have n=10 samples. Using as ...
some_math_guy's user avatar
1 vote
0 answers
32 views

How to validate the decomposed distributions?

I am fitting distributions for the time spent for three processes (i.e., pick up tools, walk to destination, install) in a system that I am trying to simulate where the original data for these ...
Haider's user avatar
  • 11
2 votes
2 answers
101 views

Showing that $E[\hat{\tau}_D] = P(n_D > 0)\tau_D$ and $\vert E[\hat{\tau}_D] - \tau_D\vert \leq \tau_D(1-\frac{N_D}{N})^n$

Consider the following double sampling scheme: We have a population of size $N$ with variable of interest $y_i$ for each $i \in \{1,\dots,N\}$, and (fixed) subpopulation $D$ of size $N_D$. Let $S$ ...
Jacobiman's user avatar
  • 147
0 votes
0 answers
33 views

Can we use Dirichlet process to simultaneously estimate the number of mixtures and component distribution of a Bernoulli mixture?

Suppose I have a random sample on a Bernoulli random variable $\{X_i\}_{i=1}^N$ generated from model $p=\sum_{k=1}^K\pi_kp_k$,where $p\equiv Pr(X=1)$ and $p_k\equiv Pr(X=1|k)$, and $\pi_k$ are the ...
ExcitedSnail's user avatar
  • 2,884
12 votes
3 answers
2k views

How do you calculate the expected value of a discrete distribution without replacement?

Say I have a set of 10 values I want to draw 3 values from, uniformly, without replacement. For instance: $$S = \{0,0,0,0,22.95,0,0,0,19.125,25.5\}$$ With replacement, this seems simple, you just add ...
brubsby's user avatar
  • 233
1 vote
2 answers
193 views

Goodness-of-fit tests for discrete distributions

I have data where only values at large x should fit to a particular distribution whose parameters I wish to determine. I want to do a goodness-of-fit test to find the value of x where the data fit to ...
user avatar
1 vote
1 answer
116 views

If hitting a target has $P = 0,3$, how many shoots to get at least one hit with a probability of $0.9$?

Cheers, I know that hitting a target has a probability of $0,3$, and I am asked to find the number $n$ of times that I have to shoot at the target to get at least one hit with a probability bigger ...
average_discrete_math_enjoyer's user avatar
1 vote
1 answer
683 views

randomized Neyman-Pearson lemma for a discrete distribution

We let $\Theta=\{0,1\}$, and $X$ be a discrete R.V with the following probability distribution: x 1 2 3 4 5 6 7 8 $f(x;0)$ 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.86 $f(x;1)$ 0.14 0.12 0.10 0.08 0.06 ...
Joe's user avatar
  • 85
1 vote
1 answer
47 views

Distibution of the number of trials required to see all possible outcomes

In the simplest case given a set of N items is the distribution for the number of draws with replacement before all items are seen? This is the case I really need. More generally what is the ...
Daniel Mahler's user avatar
0 votes
0 answers
137 views

KS test or chi square for comparing two distributions of a discrete ratio variable

I have a discrete ratio variable (length) from two samples and I'd like to know if the distributions are different or the same. In my case length is discrete because it can only take on integer values ...
An Ignorant Wanderer's user avatar
1 vote
0 answers
100 views

Approximating a countable-state (infinite) Markov model with a finite-state one

TL;DR—in a nutshell I have a countable-state Markov model (with a countably infinite number of states) in which the probability of transitioning to states $S_{i>k}$ for large $k$s are practically ...
psyguy's user avatar
  • 311
2 votes
1 answer
392 views

Central limit theorem for independent, non-identically distributed, finite discrete random variables

I am positive, that there exists a version of CLT stating that, the distribution of the sum of infinite many independent, nonidentical, finite discrete random variables, is Gaussian. I just could not ...
user2961927's user avatar