All Questions
Tagged with density or density-function
448 questions with no upvoted or accepted answers
9
votes
0
answers
509
views
Distribution/expected length of the shortest path in infinite random geometric graphs
Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
- 475
7
votes
0
answers
702
views
Spinograms vs. conditional densityplots
I have a binary response variable (hail) and multiple continuous predictor variables. My aim is to understand the linear/non-linear relationship of the predictors to the response to be able to justify ...
- 511
6
votes
1
answer
779
views
What is p(data) in image generation
In the context of image generation architectures such as VAEs or GANs (say we are using mnist digits) what do we mean by probability distribution of the data?
Just to clarify this question and make it ...
- 768
6
votes
0
answers
120
views
How can a probability densitiy be estimated based on the maximum entropy principle, with constraints in the order statistics?
Let's say we are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$.
The ...
- 1,733
6
votes
0
answers
2k
views
Cumulative distribution function for the product of two random variables
Given two random variables $x, y$, each with the probability distribution functions $p_x(x)$, $p_y(y)$, then if $z = xy$, then $p_z(z) = \int p_x(x)p_y(z/x)\frac{1}{|x|}dx $.
Is there a similar proof ...
- 71
6
votes
0
answers
2k
views
Expectation of a strictly increasing function
Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
- 860
5
votes
0
answers
458
views
Finding the probability density function of Hotelling's T-squared distribution
The following image is seen on wikipedia when searching for Hotelling's T-squared distribution
This is apparently the pdf of the Hotelling T-squared distribution at different parameters. However, I ...
5
votes
0
answers
399
views
What is the density of a markov chain when its transition probabilities have densities with respect to different measures?
I have a homogenous, discrete time Markov process, $(X_n)_{n\geq 0}$, with state space $\mathbb R_+$. Its transition probabilities have a density, $f(x_n\mid x_{n-1})$, with respect to the measure $\...
- 863
5
votes
0
answers
860
views
CDF of the ratio of two correlated $\chi^2$ random variables
It is well known that the sum of a series $m$ of squared standard independent normal random variables follows a $\chi^2$ dstribution with $m$ degrees of freedom. It is also true that the ratio of two ...
- 13.7k
5
votes
0
answers
685
views
Confusion related to Parzen window
I was going through this tutorial related to Parzen window at http://www.cs.utah.edu/~suyash/Dissertation_html/node11.html. However, I have some confusion related to Parzen window with gaussian kernel
...
- 6,847
5
votes
0
answers
159
views
How to improve estimation of a deconvolved density
I have the following problem:
Y = X + e
with
Y = Total reaction time (noisy signal)
X = selection time (signal)
e = discrimination time (noise)
I am interestend in the distribution for X and ...
- 1,411
5
votes
0
answers
201
views
Joint distribution of two distances
Suppose there are three points in 3D space, each with coordinates $A_i=(X_i,Y_i,Z_i)\leadsto \mathcal{N}(\mu_i,\tau^2\mathbb{I}_3)$. We compute the distance between the three points, e.g. $D_{ij} = \|...
- 912
4
votes
0
answers
89
views
What's the distribution of $|y-z|^2/|y-\bar{y}|^2$ for vectors with i.i.d. standard normal coordinates?
Let $y_1, y_2, \ldots, y_n$ and $z_1, z_2, \ldots, z_n$ be samples of size $n$ of a normal distribution $\mathcal{N}(0,1)$. My goal is to find the distribution of
$$\frac{\sum_{i=1}^n (y_i - z_i)^2}{\...
- 141
4
votes
0
answers
5k
views
Point process - intensity function vs probability density function
Suppose we have a point process in $\mathbb{R}$ with intensity $\lambda(x)$. Then, for a given compact set ${ S}$ we have
$$\Lambda({ S})=\int_{\rm S} \lambda(x) \, dx,$$
where $\Lambda({ S})$ is ...
- 567
4
votes
0
answers
728
views
Constructing a joint distribution from pairwise bivariate marginal distributions?
It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
- 385
4
votes
0
answers
354
views
Measure-theoretic derivation of change of variables formula for probability density functions?
Assume we have a $2$-dimensional sample space $(\Omega, B, P_\Omega)$, with $\Omega =\mathbb R^2$ with borel measure and probability measure $P$, where the axes are simply equal to random variables $...
- 2,987
4
votes
0
answers
88
views
Automatic fitting of normalization constant as a parameter in noise contrastive estimation
In the paper on Noise Contrastive Estimation, the authors define a parameterized density function $p_m^0\left(x;\alpha\right)$ to estimate the unnormalized PDF of the data, and then further define a ...
- 1,481
4
votes
0
answers
207
views
Sum of truncated Gammas and degenerate
I have a variable $X$ which I am modelling with a mixture model:
$$\begin{aligned}
(X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\
(X|B) &...
- 535
4
votes
0
answers
304
views
Testing for Normality (CDF)
I was reading an article about using the CDF to calculate the area between 2 points on the normal curve. They gave a sample of 7 for illustration purposes:
...
- 85
4
votes
0
answers
83
views
Problem involving P.D.F. containing an indicator variable
Let $X_1, X_2, \ldots$ be independently and identically distributed random variables with probability density functions: $$f(x) = \alpha \;x^{-(\alpha+1)} \; I_{(x>1)}, \; \; \alpha > 0.$$
For ...
- 579
4
votes
0
answers
2k
views
Where is the maximum bias and variance in a histogram as non-parametric density estimator?
I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me.
Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
- 755
4
votes
1
answer
258
views
Geometric construction of copula - question regarding C-volume
I am learning about copula's, using Nelsen's book, and more specifically about the geometric method of constructing copula's. The problem is replicated in the following link: http://www.stat.ubc.ca/...
- 872
4
votes
0
answers
297
views
Maximum likelihood estimation involving both probabilities and probability densities
Note: based on suggestions in the comments, I have rewritten this question. Please refer to the history for the original version.
In general my question regards how to compute likelihoods in mixed ...
- 519
4
votes
0
answers
386
views
Expectation of density ratio of two iid variables
Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio
\begin{align}
\mathbb{E}\left[\frac{f(X)}{f(Y)...
- 308
3
votes
0
answers
118
views
Conditional Distribution of Multivariate Gaussian given Linear Inequalities
Consider a multivariate Gaussian $Y\sim\mathcal{N}(\mu,\Sigma)$ of dimension $n$. For fixed $c\in\mathbb{R}^n, A\in\mathbb{R}^{m\times n}$ and $c\in\mathbb{R^m}$, what is the conditional distribution ...
- 31
3
votes
0
answers
143
views
Can I use multiple quantile regression to estimate the probability a dependant variable is above / below a certain value?
Let's say I have a dataset of characteristics of newly launched products in a retail environment, and the dependant variable Y is total $ sales in the first year of ...
- 277
3
votes
1
answer
57
views
Conditions for this functional relating densities under change of variables to exist?
Suppose I have a random variable $X$ with density function $f_X(x)$, and a continuous but non-smooth function $g$. We will also take $Y := g(X)$ to have a smooth density function $f_Y(y)$.
If $g$ had ...
- 9,680
3
votes
0
answers
177
views
PDF of the product of a Beta random variable and a Normal random variable?
If random variable $X\,\sim\,\text{Beta}(a,b)$ and $Y\,\sim\,\text{N}(\mu,\sigma^2)$, is there a closed-form solution for the pdf of their product $XY$? We assume $X$ and $Y$ are independent.
- 203
3
votes
0
answers
58
views
Deriving distribution under change of variables between spaces of unequal dimension
For a function of random variables $T:\mathbb{R}^n \mapsto \mathbb{R}^m$ Wikipedia outlines how to handle three cases:
$m = n = 1$
$m=n > 1$
$n>1 \land m=1$
There seems to be two missing cases:...
- 9,680
3
votes
0
answers
42
views
MLE is undefined for "densityless" distribution like Cantor distribution
I thought of a situation where we are given a random variable $X$ that has a Cantor "rescaled" distribution. That means that for a parameter $p>0$, $X$ has CDF $F_X(x)=C(\frac xp)$. This ...
- 435
3
votes
0
answers
84
views
Density function of a dependent sum of products of normal random variables
Say we have a random variable
$$
X = A_0 A_1 + A_0 A_2 + A_1 A_2,
$$
which consists of normally distributed independent random variables $A_0, A_1, A_2 \sim \mathcal{N}(0,1)$ with probability ...
- 31
3
votes
0
answers
242
views
Restricting Kernel Density Estimate to n-sided polygon
Given some arbitrarily distributed data, in this case data generated by two normal distributions,
$\mathcal{N_1}(\mu_1, \Sigma_1)$ and $\mathcal{N_2}(\mu_2, \Sigma_2)$ where $\mu_1 = \begin{bmatrix}0 &...
- 31
3
votes
0
answers
56
views
Integral of difference of density functions of two Continuous Random Variables goes to 0
The problem says :
Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
- 619
3
votes
0
answers
206
views
What methods are there for estimating distributions based on histograms?
I recently worked on a consulting project where a client wanted to estimate gamma and weibull distributions based purely on histograms rather than raw-data. I have never worked with problems like that ...
3
votes
0
answers
143
views
Calculating a Confidence Interval for a Proportion for a Sample of Different Size
I'm interested in a (preferably analytic) solution or approximation to the following problem:
Let $s_1$ be a sample from an unknown distribution of size $N_1$ and with proportion of successes $p_1$. ...
- 31
3
votes
1
answer
239
views
Visualizing separability / independence
I’d like to visually ‘see’ the independence of random variables. I tried plotting f(x), f(y), and f(x, y) for independent and dependent pairs of variables. However, the difference is still not ...
- 399
3
votes
1
answer
117
views
Finding expression of $n$-th derivative, when $n$ is large
For completeness, assume $C$ is an Archimedean copula with some generator function $\varphi$, which is usually assumed to have nice properties. It is known that $$ C(u_1, u_2, \ldots, u_n)=\varphi^{-1}...
- 672
3
votes
0
answers
98
views
Exponential Family Representation: Dumb question on scale parameter and whether it went to Hawaii
So going over the Hastie Tibshirani paper on GAM - it points to equation 11 as the exponential family density - but with two parameters - theta (natural parameter) and phi (scale).
https://...
- 228
3
votes
0
answers
4k
views
Finding PDF from CDF
I just got really unsure, can someone confirm/rectify? I have the CDF defined as $F(x)= \begin{cases}0, &\text{if}~x < 0,\\ 4x^2 &\text{if}~ 0 \leq x < \frac{1}{4} \\ 1-\frac{4}{3}(1-x)^...
- 39
3
votes
0
answers
96
views
PDF of function $\mbox{max}(x,f(x))$
This is from the book Python for Probability, Statistics, and Machine Learning. It's a good book for people that already have a math background, I believe. Anyway, there is something I don't ...
- 165
3
votes
0
answers
523
views
Why is the Kolmogorov–Smirnov (KS) test more popular than the Overlapping Coefficient (OVL)?
The Overlapping Coefficient (OVL) measures the common area between two Probability Density Functions (see this question for more details). Intuitively, this seems like a good way to gauge the ...
- 1,244
3
votes
0
answers
51
views
Statistics of Intervals
Question
Dear all,
Assume you ask N persons to set their personal interval of acceptance (e.g. interval ranging from 0 to 100% for the power of a speaker system) which they would rate as enjoyable. ...
- 31
3
votes
0
answers
122
views
Is the Gaussian distribution the only statistical distribution fully determined by the mean and variance?
I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by
$$ \pi_G(\xi|\mu,\sigma) = {1\...
- 724
3
votes
0
answers
137
views
Sampling from $f(x)$ given approximation $g(x)$
(After some pondering, what I really wanted to ask is how to incorporate prior information about $f$ into a sampling method - see this question.)
Suppose you want to draw samples from an (...
- 5,238
3
votes
0
answers
3k
views
Probability Density Function of a linear combination of 2 dependent random variables, when joint density is known
Let's say there are two dependent random variables $X$ and $Y$ with joint density function $f$.
What is the PDF of the weighted sum of these two variables, $Z = aX + bY$?
Thanks in advance for any ...
- 131
3
votes
0
answers
204
views
Bias of an estimated Gaussian density
I have an iid sample, $X_1,\dots,X_N \in R^d$, from a multivariate normal density with mean $\mu$ and covariance matrix $\Sigma$.
I am estimating the density $p(y) = N(y| \mu, \Sigma)$, using $\hat{...
- 3,284
3
votes
0
answers
285
views
Derivation of likelihood function for latent variable model made explicit
I am trying to make the steps deriving the likelihood function for the following latent variable model as explicit as possible: $$Y^0=X\beta + u$$ where $$u \sim NID(0,\sigma^2).$$ The observed data ...
- 502
3
votes
0
answers
67
views
Distribution with fixed mean and closest to a given distribution
I was wondering if this problem has been tackled in some way in the probability/functional analysis literature:
Given a pdf $f$ such that the expectation is zero and $\mu\in\mathbb R$, find the ...
- 786
3
votes
0
answers
62
views
Multivariate distribution for products of random variables
Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
- 93
3
votes
0
answers
938
views
Estimating time-lagged mutual information for two signal samples
This is an attempt to reproduce Moon et al. 1995, and the author's copy can be obtained through here.
As a benchmark, we estimate the time-lagged mutual information of a simple sine signal $\sin(0.02\...
- 335