All Questions
Tagged with density or density-function
1,717 questions
3
votes
1
answer
128
views
Finding the probability density function
A random variable $Z$ is obtained as follows. Let $X$ follow $U(0, 1)$, and $Y$ given $X = x$ be Bernoulli with probability of success $x$. If $Y = 1, Z$ is defined to be $X$. Otherwise, the ...
- 311
0
votes
0
answers
108
views
Approximation on Inverse Mills ratio for the normal R.V
I've come across several approximations for Mills ratio, but I haven't found any good ones for the Inverse Mills ratio. Is there any known closed-form approximation for the Inverse Mills ratio (link) ...
- 41
0
votes
0
answers
74
views
How to calculate the PDF of a modified random variable?
I have a random variable X that has a PDF defined by the following piecewise function:
$$f(x)=
\begin{cases}
x + 1 & \text{if } x \in \left[-1, 0\right]\\
-x + 1 & \text{if ...
2
votes
1
answer
92
views
CDF of $\max$ under conditions
Let,
\begin{equation}
g(\alpha,\beta) = \begin{cases}
\frac{\alpha}{\beta}, & \text{if } \alpha > \beta \\
0, & \text{if } \alpha \leq \beta
\end{cases}
\end{equation}
I want to find ...
- 233
1
vote
1
answer
109
views
Difficulties finding location and scale parameters from PDF [duplicate]
I am having difficulty finding the correct location and scale parameters for a PDF diagram that I need to validate my data.
I have already calculated the location parameter to be -1.01, but I am ...
- 13
0
votes
0
answers
60
views
Find pdf of $W = X/(X+Y)$ - Result check
I come up with this problem and I am sure if my solution is corrected.
$X, Y \sim \text{Exp}(\lambda)$. Find pdf of $W = X/(X+Y)$.
Please see my try bellow:
At Eq(17) I don't know if my result is ...
- 193
7
votes
6
answers
2k
views
Does higher variance usually mean lower probability density?
Does higher variance usually mean lower probability density? Despite the type of distribution. Thank you.
Update：
Sorry for confusion. Please allow me to clarify. If I sample the same number of data ...
- 101
12
votes
6
answers
2k
views
How to generate from this distribution without inverse in R/Python?
I am working with a distribution with the following density: $$f(x) = - \frac{(\alpha+1)^2 x^\alpha \log(\beta x)}{1-(\alpha + 1)\log(\beta)}$$ and CDF $$\mathbb{P} (X \leq x) = \int_0^x - \frac{(\...
- 197
1
vote
0
answers
138
views
Efficient estimation of conditional probability density
The formulation of the conditional density is:
$$ f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)}. $$
I need to estimate this density from data and it's prohibitively time-consuming to calculate the joint ...
- 61
0
votes
1
answer
92
views
Does a single observation from a population have the same distribution as that population?
Suppose X1 is one observation from a population with Beta(θ,1) PDF. Would X1 also have Beta(θ,1) PDF?
1
vote
0
answers
31
views
Density weighted Law of Large Numbers argument for the convergence of an expectation approximation
Given a set of IID samples $X = \{x_i\}_{i=1}^n$ assumed to be from the density $p(\cdot)$, and the function $h:\mathbb{R} \xrightarrow{}\mathbb{R}$, its expectation can be approximated as
$$\mathbb{E}...
- 11
1
vote
1
answer
453
views
Determine CDF and PDF from quantiles
I would like to determine CDF and PDF from quantiles that I have determined via quantile regression.
I have read here in the forum (Find the PDF from quantiles) that it is possible to interpolate this ...
- 11
2
votes
1
answer
103
views
Are the terms in the diffusion model equation random variables or probability density functions?
Are all terms in the first line(71) of the equation random variables or probability density functions? If they are probability density functions, is there a possibility of obtaining a value that is ...
- 21
2
votes
0
answers
36
views
The Wikipedia example of a Statistical model and its PDF
The example section of Wikipedia's article on Statistical model says:
Suppose that we have a population of children, with the ages of the children distributed uniformly, in the population. The height ...
- 121
0
votes
0
answers
226
views
Formulation of two parameter Pareto distribution
So everywhere I've looked, I have seen the two parameter Pareto distribution formulated as
$\frac{\alpha\lambda^\alpha}{x^{\alpha+1}}$. The distribution we are using in our course is $\frac{\alpha\...
- 101
0
votes
0
answers
72
views
approximating the density of the studentized range distribution
Is anyone aware of an approximation to the density function for the studentized range distribution https://en.wikipedia.org/wiki/Studentized_range_distribution ? I've found a fast approximation for ...
- 41
1
vote
0
answers
45
views
2-dimensional functions of random variables with piecewise densities
From Meyer's Introductory Probability and Statistical Applications, 2nd ed:
Suppose that the dimensions, $X$ and $Y$, of a rectangular metal plate may be considered to be independent continuous random ...
- 111
4
votes
1
answer
477
views
How to approximate non-central chisquare distribution to Poisson weighted sum of central chi-square distribution in case of non-unit variances?
According to Statistics libre texts Equation 5.9.20, a non-central chi square distribution can be approximated as sum of Poisson weighted central chi square distributions.
$\tag{1}g(y) = \sum_{k=0}^\...
- 127
2
votes
0
answers
55
views
Package for Multidimensional Density Estimation
I may be missing something obvious, but is there a python package that can reliably do density estimation of a PDF in high dimensions (e.g. 512)? I know of scipy's ...
- 21
0
votes
1
answer
351
views
create and sample from a PDF using real data
I have some highly skewed real world data, from which I want to create a probability density function which I can then use to generate random samples that mimic the original data. The data is a long ...
- 157
2
votes
1
answer
63
views
How are these simulated sample means created/plotted in R?
I had a student point out this image in Learning Statistics with Jamovi that is also in Learning Statistics with R (Page 294 in the latter). I was going to send her a reply when I noticed something in ...
- 18.5k
3
votes
2
answers
215
views
Integral of cdf of a symmetric random variable
How to compute $$\int_{-k}^{k}F(x)dx$$
where $F(x)$ is the cumulative distribution function of continuous random variable $X$ which has symmetric pdf about $x=0$ and $k>0$.
- 41
1
vote
1
answer
115
views
How do we define the pdf in the multi-variate case and compute expectations?
Apologies if this is a very simple question but trying to work through a result in a paper made me realize I missed something a bit fundamental in my undergrad probability and analysis courses. Lets ...
- 115
0
votes
1
answer
46
views
Variation of PDF [duplicate]
Suppose I have the PDF
$$
f_X(x) = \frac{12x^2}{7}
$$
with $-1 \leq x \leq 1$.
What would be the PDF of $Y = X^2$?
Would it be
$$
f_Y(y) = \frac{12x^{1/2}}{49},\quad 0 \leq y \leq 1 ?
$$
- 1
1
vote
1
answer
72
views
Conditional density following Gamma distribution
We know that if a random variable say x ~ Gamma(a, b), then its probability density function is $ \propto x^{a-1} exp^{-bx}$.
In a Bayesian hierarchical model, for example
$Z_1, \cdots, Z_n |\theta \...
- 13
1
vote
0
answers
77
views
Order Statistics - Percentile Range of Normal Mixture of Normals
Say I have draw N values from a normal distribution [$\mu_1$, $\sigma_1$]. Below are 10 sampled points compared to the normal distribution they're sampled from
I then create a normal mixture of ...
- 11
0
votes
1
answer
74
views
Maximum likelihood estimate for mixture with components using both cartesian and polar coordinates
I have a set of points (x,y) that were generated from a mixture of two components: one component uses Cartesian coordinates, and the other polar coordinates.
For example, with probability $\gamma$ I ...
- 15
0
votes
0
answers
43
views
Simulating using conditional density - is this correct?
Consider the pair of random variables $(X, Y)$ whose density is given by $$f(x, y)=yx^{y-1}e^{-y}1_{(0, \infty)}(y)1_{(0, 1)}(x)$$
I was first asked to find the density of $Y$. This was simple, that's ...
- 163
0
votes
0
answers
37
views
Two non-obviously identical random variables that can be shown to be identical via their characteristic functions
It is well known that a characteristic function (CF) is uniquely associated with a probability density function (PDF). Knowing this, I was nonetheless intrigued by a remark in a video I have been ...
- 239
3
votes
1
answer
57
views
Conditions for this functional relating densities under change of variables to exist?
Suppose I have a random variable $X$ with density function $f_X(x)$, and a continuous but non-smooth function $g$. We will also take $Y := g(X)$ to have a smooth density function $f_Y(y)$.
If $g$ had ...
- 9,680
1
vote
0
answers
45
views
Is it possible to derive pmf from a CDF with no support given?
Given a CDF of random variable, $$F_X(x)=\sum_{j=1}^x \left(\frac{1}{2}\right)^j = 1-\left(\frac{1}{2}\right)^x$$,
I want to derive pmf of this random variable $X$. But there is no information about ...
- 13
1
vote
0
answers
66
views
Density plot of probabilities [duplicate]
I have created a density plot with ggplot for the predictions of Logistic regression (probabilities).
I do not understand the y - axis. Why the range is not 0-1?
- 139
0
votes
0
answers
164
views
derivatives and distribution of a 3-dimensional copula in R
I am looking for a way to calculate in the R software, the distribution, the density and the derivatives (of order 1, 2) partial of a Gaussian copula of dimension 3.
Indeed, I have three variables (u1,...
- 1
1
vote
1
answer
63
views
How to combine two integrals containing the PDFs of a variable and its linear transform?
Original Post:
Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$.
I want to compute $Z(x) = F(x) - G(y)$.
What I have ...
- 13
6
votes
1
answer
2k
views
PDF does not integrate to 1 - where is my mistake?
I am trying to solve a question which gives me a random variable with the distribution function below
$$
F(x) = 1 - \left(\frac{\mu}{x}\right)^{2n}
$$
where $0 < \mu \le x < \infty$
I ...
- 705
2
votes
2
answers
516
views
Creating half normal probability distribution
I have come across a problem where a half normal distribution is based on a single number, namely the sum of all costs. The exact definition of the number is not important. The important think is that ...
- 487
4
votes
2
answers
223
views
Seemingly contradiction: probability density function and maximum likelihood calculations for continuous random variable
Claim 1: For continuous random variable, $P(X=x)=0$, where $x$ is a particular number.
Claim 2: When we use maximum likelihood estimation, we plug-in mean, standard deviation and data point $x$ into ...
- 365
4
votes
2
answers
659
views
Operations on Random Variables vs Distributions vs Random Samples
What is the difference between i) random variables, ii) distributions of random variables, and iii) random samples?
While trying to figure out how to average random samples from various random ...
- 2,051
1
vote
0
answers
97
views
R GLM get probability density for a GLM model
I train a GLM via
model<- glm(formula=y \sim x_1+x_2,family=Gamma
(link=log),data=...)
I now would like to get the probability density function of $f_{\theta(...
- 33
5
votes
1
answer
1k
views
Bounded density function: definition?
What is the correct definition of bounded probability density function:
$\sup_{x} f(x)<\infty$. If this is the correct definition of bounded probability density function, can you give the example ...
- 935
2
votes
1
answer
226
views
Gaussian mixture model probabilities
I'm using scipy's optimize to fit two Gaussian distributions to my data. I expected the posterior likelihood of belonging to the rightmost class to start from 0 ...
- 123
0
votes
0
answers
109
views
Storing a probability distribution without saving single values
I saw this question "Storing a probability distribution without saving single values" on stackexchange and thought it deserved a statistical answer.
Example Scenario
I could see this problem ...
- 2,051
0
votes
0
answers
38
views
Find Marginal CDF probability from PDF (2 random variables) [duplicate]
Given the following PDF of continuous 2 random variables:
$$
f_{X,Y}(x,y)=\begin{cases}
y^2 & 0\le y\le x\le 1;\newline
0 & \text{otherwise}.
\end{cases}
$$
Graph showing the ...
- 217
3
votes
1
answer
102
views
How to find the transformed density and log likelihood for this family of distributions?
Let's consider the family of transformations given by
$$g_a(Y)=\begin{cases}
\frac{e^{aY}-1}{a} & \text{ for } a\neq 0 \\
Y & \text{ for } a=0
\end{cases}$$
for $Y\in\mathbb{R}$. Analogous to ...
- 181
1
vote
0
answers
26
views
estimating derivative of a random function at a point
I have a random function $f_n(x):R\to R$. At each point $x\in R$, $f_n(x)\to f(x)$ at $1/\sqrt{n}$ rate. While $f(x)$ is smooth, the functions $f_n(x)$ for fixed $n$ are more like step functions, ...
- 21
1
vote
1
answer
40
views
Density curve inferences
I'm modeling the total revenue of sellers in a 1 year period. The distribution plot below shows quantity*price for each seller with outliers eliminated. The outliers were taken out with IQR * 1.5 ...
0
votes
0
answers
57
views
Estimation of probability with a loggamma distribution
I have an empirical observation with about 300K continuous values. I fitted these values (with disfit python library) getting a loggamma distribution:
The resulting parameters are:
c = 0....
1
vote
1
answer
321
views
Maximum likelihood estimate for mixture of different distributions
I'd like to estimate the parameters of a mixture model using MLE. The density is:
$$
f(x,y) = \mathcal{N}(x, y; \boldsymbol{\mu}, \Sigma) \cdot \alpha + \mathcal{N}(x; \mu, \sigma^2) \cdot \mathcal{U}...
- 15
2
votes
1
answer
624
views
my density/kde plots don't show what I expected
I thought I understood kde/density plots until this problem. I have a dataset with two columns, Diff and Var with 5 million rows ...
- 535
3
votes
1
answer
177
views
Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian
Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$.
I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
- 33