Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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Question about probability distributions

I've recently came across this question: ...
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1answer
38 views

Infrequent fail of the popular parameter estimators, having several beta-distributed random variables to be estimated

I have a project in which there exist $N$ Beta-distributed Random variables each of which should be estimated, having a sample for each of them. The sample domain is $\{0.1,0.3,0.5,0.7,0.9\}$ and the ...
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0answers
33 views

How do I calculate conditional PDF?

Obtain $$P(2 < Y < 3 | X = 1)$$ where the joint pdf of X and Y is $$f_{X,Y}(x,y) = (6-x-y)/8$$ where $$0 < x < 2$$ and $$2 < y < 4$$? so first, I did $$f_Y|X=1(y) = ...
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30 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
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2answers
37 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
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1answer
32 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
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1answer
21 views

CDF of the difference of two Gaussian mixtures

I have two Gaussian mixtures, $X_D$ and $X_{\overline{D}}$: $$ f(X_D) = \sum_{c=1}^m f(X_D\mid C=c)P(C=c) = \sum_{c=1}^m \phi(x-\mu-g(c))P(C=c), $$ $$ f(X_\overline{D}) = \sum_{c=1}^m ...
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1answer
70 views

Distribution of $\sin(X) *\cos(Y)$ where $X,Y$ are iid r.v., uniformly distributed on $[0, 2 \pi]$

What is the probability density of $R = \sin(X) * \cos(Y)$ where $X,Y$ are independent random variables, uniformly distributed on $[0, 2 \pi]$? I am stuck with complicated integrals, not sure if ...
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1answer
17 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
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1answer
25 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
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15 views

characteristic function problem 4

which of the following is not a characteristic function? a) 1 b) $e^{it} $ , $t \in R$ c) $\frac{1}{1-it} $, $t\in R$ d)$e^{|-t|}$, $t \in R$
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1answer
150 views

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
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1answer
31 views

How to calculate a posterior probability with a given Gaussian Mixture Model?

I'm building a GMM-based classifier in speech processing and I'm using GMM as a probabilistic scoring mechanism (therefore I don't intrinsically care about the underlying mixture components). For ...
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2answers
101 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
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1answer
32 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
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1answer
32 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
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0answers
34 views

Binomial-like distribution

Starting with $1$, for $n$ trials multiply by either $1+p$ or $1-p$, with $0 \le p< \le 1$. Does this distribution have a name? What are its properties, such as density (PDF)? It is like a skewed ...
3
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1answer
59 views

sign of the conditional expectation

I'm working on the following problem: Let $X$ be a random variable defined on $(\Omega,F,P)$ and $G$ a $\sigma$-algebra contained in $F$. Show that, if $E(|X|)<\infty$ and $E(X\mid G)$ has the ...
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1answer
57 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
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2answers
18 views

Question about finding a distribution without taking into account previous events

We have 8 prisoners, each has a probability of escaping (independently) each day of $0.4$, what is the distribution of the amount of escaping prisoners on the third day? This is the answer: the ...
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1answer
42 views

Conditional distribution of $X$ exponential given $U\leq e^{-X}$, with $U$ uniform on $(0,1)$

Let $X$ be exponentially distributed with mean $1$ and $U$ be a $U(0,1)$ random variable independent of $X$. Define $$I= \begin{cases}1,&U \leq e^{-X}\\ 0,&\text{ ...
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2answers
57 views

Distribution of a convolution.

Assume that $X_1,X_2,X_3,X_4$ are IID such that $P(X_1=0)=0.3, P(X_1=1)=0.1$ and $X_1$ has on $(0,1)$ the density $f(x)=0.6$. Calculate $P(X_1+X_2+X_3+X_4 \leq 1).$ My work so far. It seems that ...
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3answers
33 views

How can $e^X \thicksim \mathrm{logN}(\mu, \sigma^2)$ given $X \thicksim N(\mu, \sigma^2)$ when they have different support?

According to Wikipedia (page about lognormal distribution), if $X \thicksim N(\mu, \sigma^2)$ then $Y=e^X \thicksim \mathrm{logN}(\mu, \sigma^2)$. But the support of $\mathrm{logN}$ is just ...
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1answer
23 views

Recovering density parameters from distribution function

Let $X$ be a random variable with probability density function $g(x;\theta_1,\theta_2)$, where $g$ is parameterized by two real numbers $\theta_1$ and $\theta_2$. I'd like to specify that $$ P(a \leq ...
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83 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
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2answers
58 views

Random increment through a probability distribution function

To Clarify i am trying to generate a random variable from a gamma pdf If $\Delta X$ indicates a random increment and it is said that $\Delta X$ follows a Gamma distribution. What would that mean ...
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0answers
52 views

Help Integrating $I=\int\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$

I am trying to integrate the following function involving the Normal CDF ($\Phi$). I actually need the definite integral $$\int^b_a\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$$ for $q+ra,q+rb >0$ but ...
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34 views

Can I compute marginal distribution this way?

I have posted the same question in the Internet another website. But I did not get the answer replies. I only can come here to have a try. The math statement I put here may not be correct. You can ...
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1answer
38 views

Sum of two truncated normaly distributed variables

Let $X$ and $Y$ be two variables which are truncated normally distributed above zero (that is $X$ and $Y$ have the lower truncation point zero, their values are bounded above zero). Is $X+Y$ truncated ...
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1answer
21 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
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3answers
38 views

Probability of Punctures for a group of cyclists

The matter of the probability of punctures occurring cropped up during a ride yesterday with a friend. His view is this, (As we can't let a subject drop.... ;-) ) "Eric, There must be more chance ...
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36 views

Help with textbook formula

In Bishop - Pattern Recognition and Machine Learning, Section 1, I do not fully understand Formula (1.65). Although it's not stated explicitly, I assume that I is the identity matrix with the ...
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2answers
34 views

A smooth function satisfying these functional constraints

I am looking for any function on a square $$f:[-1,1]\times [-1,1] \rightarrow [0,1]$$ with the following properties: The function $f$ is as smooth as possible, e.g. differentiable almost everywhere. ...
2
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1answer
72 views

Transformation theorem

Given $X_1$ is $\Gamma(\alpha,1)$ distributed and $X_2$ is $\Gamma(\beta,1)$ distributed and set $$Y=\frac{X_1}{X_1+X_2}.$$ The task is to show that $Y$ is $\operatorname{Beta}(\alpha,\beta)$ ...
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1answer
50 views

Probability of an event that occur first of a joint uniform distribution

A man and a woman agree to meet at a certain location about 12:30 P.M. If the man arrives at a time uniformly distributed between 12:15 and 12:45, and if the woman independently arrives at a time ...
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16 views

4th order correlations of a delta-correlated random process

Say I have a complex random variable A(z) that is $\delta$-correlated, i.e. I have: $ \begin{align}\langle A(z) \rangle &= 0 \\ \langle A(z) A^*(z') \rangle &= \delta(z-z') \end{align} $ ...
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41 views

Topology of statistical manifolds

I am currently working with statistical manifolds. Roughly, a statistical manifold is a set of distribution parametrized by a set of parameters. However i have trouble finding more precise definition. ...
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1answer
20 views

The distributions of incomes in two cities follow the two Pareto type pdfs. Find P(X<Y)

The distributions of incomes in two cities follow the two Pareto type pdfs $$f(x)= \frac{2}{x^3}, 1 < x < \infty.$$ $$g(y) = \frac{3}{y^4}, 1<y<\infty.$$ Here one unit represents ...
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1answer
28 views

What is the time between groups of events when single events have a Poisson distribution?

I'll ask this with a concrete example to be clear. Let's say I have a Poisson process that tends to produce one event every two minutes. Then the probability of getting an event in a given minute is ...
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0answers
16 views

To what set do measured values belong?

This question is more conceptual than practical. It seems that when we apply mathematics to measured values, we treat them like real numbers. When measured values take error into account using ± ...
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3answers
35 views

Creating a PDF for a discrete random variable with a countably infinite set of values?

I am unsure how to transition from discrete random variables with a finite set of values to ones with a countably infinite set of values. The question that spawned this problem: A bucket has two ...
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1answer
37 views

How to reconstruct distribution from the generating function

Suppose $$F(x)=\sum_n p(n)x^n$$ is a generating function, and we have the expression for $F(x)$ explicitly. Then how we can get the expression for $p(n)$ from this generating function?
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34 views

difference between limiting and special case

In Mathematics and Statistics we see generalized distributions having a number of parameters. varying the values of these parameters we get special or limiting distributions of the generalized ...
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1answer
42 views

Generate random number according to any equation

So I'm after a random number generator where the probabilities of a number occurring in some range is matched to some function. Only really looking at functions with nice integrals (for simplicity ...
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51 views

If we know X is a Poisson binomial random variable what can we say about mX?

Suppose that X is sum of m independent Bernoulli random variables that are not necessarily identically distributed, and thus it has Poisson binomial distribution. Is mX also a Poisson binomial random ...
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38 views

Probability ( Random Variable ).

Let $$p_X(x) = \begin{cases}\frac{x}{15},& x\in\{1,2,3,4,5\}\\ 0,& \text{otherwise}\end{cases}$$ be the probability mass function of $X$. We need to find $$\mathbb ...
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1answer
45 views

Expectation on second moment which involves linearity

I have a small problem regarding to expectation on second moment. It would be lovely if you guys can give me a hand. The amount of a claim that a car insurance company pays out follows an ...
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1answer
67 views

Probability distribution of bored people

5 people are arranged in a row, a person is talkative with a probability of $p$ and silent with a probability of $1-p$, each is independent. A person is bored if he's talkative and sits between two ...
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33 views

Expectation of $\min(X, c)$ for $X$ truncated r.v. and $c$ constant

I have a random variable $X$ and a constant $c\geq 0$. I define the r.v. $Y = \min(X, c)$ and I want to calculate $E[Y]$. I have seen different posts on similar topics, so I am trying to pull all ...
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Is there a map that maps the mean vector and the variance matrix of a multivariate lognormal to its location vector and diffusion matrix?

Suppose ${\xi} \sim logNormal_d ({\mu},{\Sigma})$, where $\mu$ is a d-dimensional vector (called location vector) and $\Sigma$ is a $d \times d$ symmetric positive definite matrix (called diffusion ...