Questions tagged [discrete-distributions]
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51
questions
7
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2answers
98 views
Let $X,X_1,X_2,X_3,…$ be positive integer random variables. Show that $X_n \overset{d}{\to} X$ implies $\lim_{n\to\infty} P(X_n=k) = P(X=k)$
Question
Let $X,X_1,X_2,X_3,...$ be positive integer random variables. Show that $X_n \overset{d}{\to} X$ implies $\lim_{n\to\infty} P(X_n=k) = P(X=k)$.
The $\overset{d}{\to}$ denotes convergence in ...
7
votes
4answers
610 views
How to interpret sum of two random variables that cross domains?
suppose we have two discrete random variables:
$X: \{$6 sided dice rolls$\}$ $\rightarrow \{1..6\}$ (following uniform distribution)
$Y: \{$coin flips$\}$ $\rightarrow \{0,1\}$ (following uniform ...
5
votes
4answers
158 views
If $X$ and $Y$ have the same marginal distribution, then do they have to have the same conditional distribution?
Suppose $X$ and $Y$ are two random variables that have the same distribution. Does
$$P[X \leq t \mid Y=a]$$
be necessarily equal to $$\;\ P[Y \leq t \mid X=a]?$$
Note that if $X$ and $Y$ are bivariate ...
4
votes
3answers
106 views
Distribution of X+U when X is a discrete and U is a continous random variable
Suppose $X$ and $U$ are independent random variables. $X$ is a discrete uniform variable and $U$ is a continuous uniform $[0,1]$ variable. What is the value of $\mathbb P(X+U\leq y)$, where $y$ is a ...
4
votes
1answer
45 views
Finding the mode given the probability of occurence
When a teacher asks a question, a student has a probability of 0.4 of being asked. Assume the occurrence is independent.
What is the mode of the number of questions raised by the teacher it takes for ...
4
votes
1answer
84 views
The meaning of a parameterization of the logarithmic distribution
In calculus one learns that
$$
p + \frac{p^2} 2 + \frac{p^3} 3 + \frac{p^4} 4 + \cdots = -\log(1-p). \tag 1
$$
Thus a discrete probability distribution on the set $\{1,2,3,\ldots\}$ is given by
$$
\Pr(...
3
votes
2answers
144 views
Sampling distribution of the mean of the discrete-power law distribution
For a certain problem I wish to generate random integers $k$ so that their distribution follows $p_k \sim k^{-\alpha}$ for $k \geq k_{\text{min}}$, $k_{\text{min}} > 0$. I am following the ...
3
votes
2answers
153 views
Do financial return series have a probability mass function (pmf)?
Stock returns, computed from stock prices as $r_t = \ln (p_{t}) - \ln (p_{t-1})$, are real-valued and unbounded giving the impression that they are continuous random variables. But aren't they ...
3
votes
1answer
81 views
Sufficient Statistics and Discrete Distributions
Let $X_1, \ldots, X_n$ be a random sample of size $n$ from the following distribution:
$$f(x;\theta) = \left\{\begin{array}{ccc} \frac{1 - \theta}{6} & , & x = 1 \\ \frac{1 + \theta}{6} & ,...
3
votes
1answer
77 views
Conditional expectaction with probabilities for a sum of independent random variable
I have a r.v $S_N$ built as a sum of Bernoulli with parameter $p$. So $S_N = X_1 + X_2 + \ldots + X_N$. There is a second variable N, such that $N \sim Poisson(\lambda) $.
I have to compute:
$P(S_N=0)...
3
votes
1answer
32 views
Looking for a discrete distribution with a specific mean-variance relationship
Say we have some counts $Y$ for which the mean-variance relationship is
$$
Var[Y] = \alpha E[Y] + \beta E[Y]^2.
$$
From this, we can say that:
If $\alpha = 1$ and $\beta = 0$, then $Y$ can be ...
2
votes
2answers
31 views
What is the probability that the third largest value of 39 rolls with a 9-sided dice is 8?
If I have 39d9, that is 39 rolls with a 9-sided (fair) dice, what is the probability that the third-largest value of all the rolls is an 8?
I know there are some specific situations where this is ...
2
votes
1answer
33 views
Convolution of probability mass functions (3 non-parametric distributions)
I am familiar with the convolution of probability mass function when it involves two random variables, but I get a little confused when there's a third one.
I have to find the probability mass ...
2
votes
1answer
63 views
Expression for Probability of Being Between Two Poisson Random Variables?
I have two independent Poisson random variables $A \sim \text{Poisson}(\lambda_A)$ and $B \sim \text{Poisson}(\lambda_B)$. For a fixed given integer $k$, I'd like to determine
$$P(A < k \leq A + B)....
2
votes
1answer
25 views
Does $ \sum_a P(a|b,c)P(d|a) = P(d|b,c)$?
I saw it applied in a textbook once, but can't seem to figure out why or if it holds.
2
votes
1answer
52 views
Parameterized probability distribution for finite, discrete values?
Sorry if I don't have the right terminology for asking this question in a good way ...
I'm curios if there is an established distribution function for the following case:
I have 20 different options, ...
2
votes
2answers
115 views
Is there a discrete distribution I can use for sampling in R?
Firstly, I don't have a stats background, so please accept my apologies for any errors or misunderstandings in the question below.
I'm trying to use R to draw values from a discrete probability ...
2
votes
1answer
49 views
Intuition for expectation of discrete random variable that takes positive integers
If $X$ is a discrete random variable that takes values on the positive integers, it is true that
$$E(X) = \sum_{k=1}^{\infty} P(X \ge k)\;.$$
I know how to prove this (by expressing the summand as a ...
2
votes
1answer
58 views
Mathematical Statistics (Wackerly, Mendenhall, Scheaffer): Problem 4.98: Mixed Probability Distribution
Problem Statement. The duration $Y$ of long-distance telephone calls (in minutes) monitored by a station
is a random variable with the properties that
$$P(Y=3)=0.2\qquad\text{and}\qquad P(Y=6)=0.1.$$
...
2
votes
0answers
23 views
Maximum entropy discrete distributions with specified mean
Consider a discrete distribution on {1, ...,n}, with mean given as $m$, what is the maximum entropy distribution?
I know it takes the form $p_{X}(k)=ar^{k}$ and is a geometric distribution when n is ...
2
votes
2answers
28 views
Number of draws required to draw a given number of special objects in SRSWOR
Suppose we have a set of $N$ objects with $K$ special objects of interest. We draw from our set of objects using simple random sampling without replacement (SRSWOR), and we are interested in ...
2
votes
0answers
46 views
Measure how good is the discrete approximation of continuous random process?
If continuous random random process approximated with discrete version - how to measure how good is the approximation?
One possible way is to compute and compare the moments. But for some probability ...
1
vote
1answer
27 views
What is the name and formalism of this discrete distribution? [closed]
I am searching the name of something similar to a binomial distribution, but with individual probabilities (P(1) to P(N)).
I calculated (brute-forced with a script) the probability of k positive ...
1
vote
1answer
27 views
Can we always write a random variable as conditional expectation plus error?
Consider the random variables $Y,X$. I believe that we can always write
$$
Y=E(Y|X)+\epsilon
$$
with $E(\epsilon|X)=0$.
Question: Is the above true regardless whether $Y$ is a discrete or continuous ...
1
vote
1answer
53 views
How to plot density for repeated k-fold cross validation?
Long story short, I conducted regression using repeated k-fold cross validation. While messing around I decided to plot the density of the R-squared distribution for the resampling. Obviously there ...
1
vote
1answer
118 views
intuition behind Brier score
Assume that we have some count data $x_{1}, \dots, x_{n}$, which take values $\{1, \dots, m\}$
and we have some estimator of the probability mass function, $\hat{\mathbf{p}} = (\hat{p}_{1}, \dots, \...
1
vote
2answers
111 views
How to know whether a zero-inflated model is the way to go? Both poisson and negative binomial do not fit my count data
I have a dataset with count data as response variable ranging from 0-5 (number of chicks fledged). I intend to carry out a GLMM and need to know which distribution my data follows. I used the descdist(...
1
vote
1answer
22 views
Error on the mean for discrete histogram
Suppose I have a discrete histogram, in this case the number of votes for each rating (1-5 stars) for a product.
I can calculate the sample mean easily enough, and get an average score for this ...
1
vote
1answer
33 views
Can the pdf of the difference of two independent random variables be found out using their Probability Generation Functions?
I recently learned of probability generation functions and that the sum of two independent random variables can be found out by multiplying the PGFs. I wanted to know if anything similar can be done ...
1
vote
2answers
50 views
Estimation of Discrete random variables
Suppose you have a discrete random variable $X_1$ with known probability mass function. I guess that choosing a variable drawn from the same pmf would be the best way to guess $X_1$ assuming all ...
1
vote
1answer
240 views
How to solve for the minimum KL Divergence when the distribution is discrete?
Say we have a simple case of $p(x,y)$ is a 3x3 matrix:
$$\begin{bmatrix}
1/6 & 0 & 0 \\
1/6 & 3/6 & 0 \\
0 & 0 & 1/6
\end{bmatrix}$$
And $q(x,y)=...
1
vote
1answer
27 views
Categorical sampling without instantiating probability vector
I want to sample from a discrete distribution with probability vector $p \in \mathbb R^n$, where $n$ is large.
Suppose that $p_i = f_i / Z$, where $Z$ is a normalization constant. I can compute the ...
1
vote
0answers
15 views
How to prove or disprove that a complete sufficient statistic exists?
We have a discrete random variable which takes values with probabilities $p, q, p+q$ and $r$. I want to construct a complete sufficient statistic based on a single observation from this distribution, ...
1
vote
1answer
27 views
What are the limmitations of reparametrization gradients for discrete random variables? (Gumbel-softmax)
We know that one approach for re-parametrizing gradients for variational inference is taking the Gumbel-softmax estimator proposed in [1] and [2].
In [3], that is a talk on SVI, D. Blei at around 29:...
1
vote
1answer
115 views
Using one-sample Kolmogorov-Smirnov test impementation for comparing two samples with purely discrete CDF
In one-sample Kolmogorov-Smirnov test one tests whether sample $X = (X_1,\dots,X_n)$ come from theoretical distribution by computing $D_n = \sup_{x} |F_n(x) - F(x)|$, where $F_n(x)$ denotes the ...
0
votes
1answer
37 views
Textbook probability question - is this unrelated data to throw you off?
4.24 from Newbold 8th Edition:
The probability of a team winning a game is 0.45. What is the probability that the team will:
a) win 2 games out of 5;
b) win 10 games out of 25?
If the ...
0
votes
0answers
19 views
MLE and binary variables
Suppose $z$ and $y$ are discrete random variables taking values 0 or 1. The distribution of $z$ and $y$ is given by $$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
votes
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 ...
0
votes
0answers
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
votes
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 ...
