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Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions. Use this tag along with the tags (probability), (probability-theory) or (statistics).

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modification of Cochran's theorem

Assume we have $X_{1}, \dots, X_{n}$ are iid Standard Normal random variables. Next, let $Q = X^{T}AX$, where $A$ is a matrix with rank $r < n$. Next, assume we have $$ Q = X^{T}B_{1}X + \dots + X^{...
ABK's user avatar
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25 views

Probability of alien civilizations reaching Earth by combining the Drake Equation with Random Walks.

Goal To estimate the probability of alien civilizations reaching Earth by combining the Drake Equation with random walk theory. If the aliens reach a sphere of radius 50 light-years around Earth (due ...
vengy's user avatar
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Characteristic function and Student's t-distribution; a problem

I'm not sure if it is a mistake in the text or if it is actually meant to be like this, but I'm reading An Intermediate Course in Probability by Gut, and in the chapter on moment generating function (...
psie's user avatar
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Calculate the normal cone of a set of increasing functions

Let $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ increasing}\} $. We endow it with $L_2$ norm, and thus $X \subset L_2([0,1])$. We can show that the set $X$ is closed and thus compact, and also ...
PeterL's user avatar
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2 votes
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52 views

Derivation of the expected minimum distance between uniform random variables on the unit interval

Suppose we have $n$ iid random variables, $U_1,...,U_n\sim\text{Unif}(0,1)$. What is the expected minimum distance between any two of these random points? Now, I know this has already been asked and ...
aaaaaaaaaaaaaaaaaanon's user avatar
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3 views

Central Limit Theorem for Bounded Random Vectors with Dependency Graphs

I am familiar with the following Central Limit Theorem (CLT) result for a family of bounded random variables with a dependency graph structure (Paper Link): Let $\{Y_{1}, \ldots, Y_{d}\}$ be a family ...
Bhisham's user avatar
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35 views

conditional probability given continuous random variable

I am confused with conditional probability. Assume $X$ and $Y$ are continuous random variables. Let $f_{X}(y|x)$ ba a conditional density. Then if we compute $\int_{-\infty}^{y}f_{X}(t|x)dt$ isn't it $...
LrM's user avatar
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Finding upper bound on $\mathbb{P}(|\frac{X}{n}-p|>a)$ for $X\sim Bin(n,p)$

In Jacod & Protter's "Probability Essentials" exercise 5.14, I found the following question For $X\sim Bin(n,p)$, $p,a>0$, $n\in\mathbb{N}$, show that \begin{equation} \mathbb{P}(|\...
Leoncino's user avatar
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Resampling of binary sequence and probability of total successes [closed]

We are given a binary population with $s$ ones and length $N$ (therefore $N-s$ zeros), and sampling, without replacement, $r$ elements ($0 < r < N$) from it, yielding a sample with $m$ ones of ...
vonPetrushev's user avatar
1 vote
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Quant Probability Question -- Exponential Distribution

Consider the following problem from QuantQuestions.io: Two machines independently manufacture Bakugan and Beyblade toys. The time it takes each machine to produce a toy is Exp(1) distributed. The ...
Eli Yablon's user avatar
1 vote
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14 views

Conditions for a Random Variable to Satisfy a Probability Bound on Boundary Points.

Let $X$ be a random variable supported on $\mathcal{X}\subset\mathbb{R}^{d}$, and let $\mathcal{X}$ be compact. Consider $ f $ as the probability density of $ X $. My question is: What conditions ...
Diego Fonseca's user avatar
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11 views

Connection between regularity of homogeneous random field in $R^2$ and absolute continuity. Has this been generalized to the $n$-dimensional case?

What I am talking about is a rather old mathematical paper, published in Russian, from 1957. The name of the paper is “О линейном экстраполировании дискретного однородного случайного пoля”. I cannot ...
S-F's user avatar
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2 answers
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Confusion in using applying variance formula

Consider the question An elevator’s weight capacity is 1000 pounds. Three men and three women are riding the elevator. Adult male weight is normally distributed with mean 172 pounds and standard ...
Richard Gene's user avatar
1 vote
0 answers
9 views

Is it possible to apply kolmogorov smirnov for class interval data?

Let`s suppose i have the data bellow, how should i calculate the ECDF's and the how should i calculate the D critical value? I'm trying to follow the steps from https://www.ime.unicamp.br/~dias/...
chunli's user avatar
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1 answer
31 views

Is there a distribution of values such that removing values decreases the mean?

I am working with some data in which I find that, as more and more samples are collected, the average value of the samples decreases over time. Originally I thought that there was just some physical ...
Boone's user avatar
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Is this step in the asymptotic analysis of a certain physical system ok?

In analyzing the thermodynamic limit of a certain system in statistical mechanics, I've encountered the following situation: $f_n$ is a sequence of probability density functions on the real line with ...
joshphysics's user avatar
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1 vote
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19 views

How to calculate the value at risk for a spherical distribution?

I want to calculate for a spherical random variable $Y \sim S_d({\psi})$ (here $\psi$ is the characteristic generator) the value of $\mathrm{VaR}_\alpha(Y_1)$, i.e. the value at risk of the first ...
julian2000P's user avatar
2 votes
0 answers
46 views

Finding the optimal distribution that maximises

Consider two discrete random variables $X$ and $Y$. Let $Q$ be a distribution over $X, W$ be a conditional distribution $Y$ given $X, U$ be a conditional distribution $X$ given $Y$, and $s$ and $r$ be ...
pmoi's user avatar
  • 97
2 votes
1 answer
38 views

Definition of mixture of two distributions

What is the formal definition of a mixture of distributions? Usually one says that we flip a coin, and then choose a distribution out of the two to follow. Is it formally correct, then, to say that a ...
xyz's user avatar
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2 votes
1 answer
79 views

Estimation of a gamma function-like integral

A random variable $X$ has a pdf: $$f(x) = \frac{1}{k!} \cdot x^k \cdot e^{-x}$$ Prove that $$P(0<X<2\cdot(k+1)) > \frac{k}{k+1}$$ There are no conditions about $k$, so it can be non-integer. ...
Disciple's user avatar
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1 vote
0 answers
42 views

Distribution of $T_j$ [closed]

I have a binary matrix (All elements are $0$ or $1$) $A \in \mathcal{R}^{K\times L} $ with some rules. The rules are as follows: Each row must have $R$ $1$s. Each column must have $C$ $1$s. $(C = \...
Jake Jeong's user avatar
1 vote
2 answers
53 views

Discretized Distributions on Rationals?

Consider the measure space $(\mathbb{Q}, 2^{\mathbb{Q}}, \nu)$, with $\nu$ being the counting measure. The space is then $\sigma$-finite. Is there any attempts made to define analogues of continuous ...
温泽海's user avatar
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-1 votes
2 answers
37 views

Are $\mu_{\hat{p}}$ and $\sigma_{\hat{p}}$ considered parameters or statistics?

Is $\mu_{\hat{p}}$ (the mean of the sampling distribution of $\hat{p}$) and $\sigma_{\hat{p}}$ (the standard deviation of the sampling distribution of $\hat{p}$) considered parameters or statistics, ...
36n's user avatar
  • 15
-4 votes
1 answer
41 views

how to solve this joint probability distribution question? [closed]

Given the joint probability density function for $(X,Y)$: $$f(x,y) = \begin{cases} \frac{x+y}{2} & \text{if } x>0,y>0,3x+y<3 \\ 0 & \text{otherwise} \end{cases}$$ How ...
Prasanna Kotyal's user avatar
0 votes
1 answer
23 views

Looking for correct keyword for a "time series" however each random variable has different probability distribution.

precisely I am looking for the following. Given a set of random variables $X_0,X_i,...X_n$. Each RV has its own probability distribution. Additionally, RV with larger index is dependent on the RV with ...
Chen's user avatar
  • 1
0 votes
0 answers
97 views

Multivariate Hypergeometric Opening Hand Calculator

So for part 2 of creating a Yugioh Card Game opening hand calculator. I want to find a way to a way to calculate the probability to draw 3+ cards of a certain card type. In Part 1 I already ...
RoMeCAESAR's user avatar
3 votes
0 answers
65 views

Solution of equation with unknown under the integral

I have a problem which I have reduced to solving the following equation for the unknown $r_0$: $$ 1/2 = \int_0^D f(r)p(r,r_0)dr $$ where $D \in \mathbb{R}$, and $f$ is continuous density function. $p(...
Ollie's user avatar
  • 103
0 votes
1 answer
33 views

How to find (power) distribution parameters

There is a set of values (vector, array), it is expected that they have a distribution of $Ax^D$ type, where D < 0, and $x_i$ are in the range [m;M] (range [0;∞) is impossible - the integral ...
Harry's user avatar
  • 121
-1 votes
0 answers
34 views

Distribution of two combined ML models

Due to the complexity of the problem, the problem was divided into two models: a stationary model and a model that corrects the stationary model for temporal effects, i.e. $X = X_{stat} + X_{time}$ ...
xbc68's user avatar
  • 1
-1 votes
0 answers
20 views

Wishart distribution with non-full rank linear transformation [closed]

I have a Wishart variable $X \sim W_p(V,m)$ of dimensions $p \times p$ and a linear transformation $C$ of dimensions $q \times p$ and rank $q-1$ with $q < p$. Is there some way to find the ...
Robert Morgan's user avatar
3 votes
1 answer
261 views

How to Define Higher-Order Terms Analogous to Expectation and Variance in Probability Theory?

Let $X$ be a random variable, with its expectation denoted as $\mu^{(1)} := E[X]$. Correspondingly, its variance is defined as $\mu^{(2)} := \text{Var}(X) = E[(X - \mu^{(1)})^2]$. We have come up with ...
VerMoriarty's user avatar
0 votes
0 answers
18 views

A Calculation of Limiting Distribution by Fourier Transform

I am studying asymptotic formula for the periodic orbit in suspension of shift of finite type. The attached picture shows a part that I don't quite understand Here $\lambda(\tau)$ is the quantity ...
JNF's user avatar
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1 vote
0 answers
35 views

Simple proof (avoiding Gamma function) for a Gaussian lower bound?

Let $g \sim N(0, I_n)$ be a standard multivariate Gaussian vector in $\mathbb{R}^n$. It can be shown via use of Gamma function identities and inequalities that $$ \sqrt{\frac{n}{n+1}} \leq \mathbb{E}\...
Drew Brady's user avatar
  • 3,774
1 vote
0 answers
18 views

How restrictive is the single-crossing property for a mean-preserving spread?

I am trying to prove some results based on a mean-preserving spread of a distribution, and they basically depend on the distributions satisfying a single-crossing property, like the figure here: ...
radaic's user avatar
  • 11
0 votes
1 answer
37 views

An inequality for a bisected "shifted quadrant" under a continuous symmetric bivariate distribution?

Suppose a bivariate probability distribution is continuous and has circular symmetry about the origin; i.e., the lines of constant density are concentric circles centered on the origin. Now consider ...
r.e.s.'s user avatar
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-2 votes
0 answers
28 views

Kolmogorov bound for comparison of Random Vector Projections on a Sphere [closed]

Let $n$ be a fixed integer and $X$ be a random vector in $\sqrt{n} S^{n-1}$ (the $\sqrt n$-radius sphere in $\mathbf{R}^{n}$) with a density $f$ which satisfies the following property: $ \forall x \in ...
Yass1's user avatar
  • 1
0 votes
1 answer
27 views

posterior covariance of a gaussian process

I'm currently studying gaussian processes. In this framework we build "stochastic" functions f for instance $\mathbb{R}^N\mapsto\mathbb{R}$. If I've got $M$ input $X_i\in\mathbb{R}^N$ ...
Oersted's user avatar
  • 171
2 votes
2 answers
103 views

The distribution of $XY+(1-X)(1-Y)$ for $X,Y$ sampled uniformly from [0,1]

Let $X,Y$ be sampled uniformly from the interval $[0,1]$ and $Z=XY+(1-X)(1-Y)$. I would like to know the exact distribution of $Z$. I conjecture it should be uniform as well, but was not able to prove ...
user50394's user avatar
  • 429
1 vote
1 answer
116 views

Card Game Opening Hand Calculator with subgroups

So I'm trying to create a Yugioh Card Game opening hand calculator in a google spreadsheet. This can also be seen as drawing balls from an urn without replacement, but every ball has multiple ...
RoMeCAESAR's user avatar
1 vote
0 answers
51 views

Finding the general convolution of probability function with hypergeometric PDFs.

I am trying to find the generalized convolution of this PDF distribution. $$f(n, \sigma; v)= \frac{2^{\frac{3}{2}-\frac{n}{2}} \sigma ^{-n-1} v^\frac{n - 1}{2} K_{\frac{ n - 1}{2}}\left(\frac{v}{\...
DysonSphere's user avatar
0 votes
0 answers
12 views

Removing vertices from rooted tree to make it balanced

The question says, what is the least number of vertices that must be deleted from T to yield a balanced tree. The correct answer is 1. But how, i see the graph is already balanced and doesn’t need ...
kic srx's user avatar
  • 11
0 votes
1 answer
28 views

Measuring departure between the posterior predictive distribution and the true data generating distribution

Suppose, I am trying to measure the departure between the posterior predictive distribution and the data generating distribution. So, in this case, assume that there is a single observation $$X \sim N\...
Maths Freak's user avatar
2 votes
1 answer
56 views

What are the restrictions on covariance matrices of nonnegative random variables?

If $M \in \mathbb R^{n \times n}$ is the covariance matrix of nonnegative random variables $X_1, \dotsc, X_n$ with $\mathbb E[X_1] = \dotsb = \mathbb E[X_n] = 1$, i.e. $M_{ij} = \mathbb E[X_i X_j]-1$, ...
Greg Rosenthal's user avatar
1 vote
0 answers
28 views

Identically distributed then conditionally identically distributed

This seems really easy, but I can't exactly formulate it. Let $X$, $Y_1$, $Y_2$ all be identically distributed, but $Y_1$ and $Y_2$ are dependent. I want to know whether the conditional distribution ...
Joshua Woo's user avatar
  • 1,203
1 vote
1 answer
38 views

Doubts on "An Intensive Introduction to Cryptography" exercise about Shannon's entropy

I was going through the exercises in An Intensive Introduction to Cryptography (see full PDF here), and in particular, I had some doubts on Exercise 0.12 (found on page 42). Here is the relevant ...
chirpyboat73's user avatar
0 votes
0 answers
24 views

Understanding of Probability density function and Radon–Nikodym derivative

In the measure-theoretic formalization of probability theory: A random variable $X$ is defined as a measurable function $X$ from a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ to a ...
MathAccount12's user avatar
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0 answers
24 views

Understanding the Inequality Involving the Probability of Algorithmic Error and Expectation

I am trying to understand the following two inequalities involving a random symmetric matrix $\Theta$ and an algorithm $B$: $ P(B, \Theta) \geq 0.5 \cdot \mathbb{E}\left[\mathbb{I}_{K,L \text{ is not ...
Alan Bakar's user avatar
0 votes
0 answers
12 views

Density belongs to the Kullback-Leibler support of prior distribution, proof verification.

Suppose that we have some data generated from the a density $f_{0}$ with unknown true parameter $p_{0}$. Then $f$ is the density with parameter $p$ that we use to infer the true density $f_{0}$ or ...
Jonathan1234's user avatar
  • 1,083
1 vote
0 answers
29 views

Moments of two probability distributions match. Can the distributions have support differing by an interval?

Let me define the support of a probability distribution $f$ as the set of all locations $x$ where $f(x)>0$. I'll denote it $S(f)$. Suppose the distributions $f$ and $g$ have matching moments. Can ...
user196574's user avatar
  • 1,846
1 vote
1 answer
64 views

Distribution of a product of random variables

I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF $$F_X(x) = \begin{cases} F_X^1(x) & x \in (-\infty, a_1)\\ F_X^2(x) & x \in [a_1, a_2)\\ F_X^3(x) & x \...
rkim's user avatar
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