# Questions tagged [discrete-distributions]

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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 $x$. ...
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### 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), ...
1 vote
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 ...
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 ...
• 174
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### 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 $k<n$. ...
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1 vote
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### 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 $N \gg 6$ ...
• 153
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?
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 ...
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• 1,216
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### 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 ...
• 661
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### 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 ...
1 vote
127 views

### Proving independence of discrete variables and the product of them

Given that P(A) and P (B) are independent and $P(A=1)=\frac{1}{2}$ $P(A=-1)=\frac{1}{2}$ $P(B=1)=\frac{1}{2}$ $P(B=-1)=\frac{1}{2}$ There is a random variable $C = A \cdot B$, C is ...
1 vote
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 ...
1 vote
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?
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### 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 ...
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 ...
• 9,137
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 ...
• 110
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 ...
1 vote
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 ...
• 113
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 ...
1 vote
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 ...
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### 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 $S$ ...
• 147
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### 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 ...
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### 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 ...
• 233
1 vote
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 ...
1 vote
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 ...
1 vote
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 ...
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1 vote
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 ...
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 ...
1 vote
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 ...
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