Questions tagged [pdf]

Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

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29 views

What's the purpose of using “algorithm = ball_tree” in sklearn.neighbors.KernelDensity?

In sklearn.neighbors.KernelDensity,there is a parameter "algorithm = ball_tree". What is its specific role in KDE? ...
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Estimation of the conditional density functions

Suppose we have a random vector $x=[x_1, x_2, ..., x_{10}]\in\mathbb{R}^{10}$. Obviously, $x_1$, $x_2$, ..., $x_{10}$ are random variables theirselves. I have 500 observed samples of $x$ which I am ...
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90 views

Relationship between a logistic decision function and Gaussian Noise

Imagine an experiment, in which an observer has to discriminate between two stimulus categories at different contrast levels $|x|$. As $|x|$ becomes lower, the observer will be more prone to making ...
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80 views

Probability function for difference between two i.i.d. Exponential r.v.s

My answer is completely off. Can you please tell me where did my logic go wrong. Donald Trump and Tori Black are to meet at a specific time and both will be late by $ \sim Exponential(\lambda), i.i.d. ...
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28 views

What type of distribution for epsilon? can I generate observations for epsilon?

$X$ has zero truncated poission distribution ($\lambda$) then pmf is $f(x)=\frac{\lambda^x}{(e^\lambda-1)x!}$ and the probability generating function is $\phi_X(s)=\frac{e^{\lambda s}-1}{e^\lambda-1}$...
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Simple function of two random variables, $Z = \frac{Y}{X}$ where Y, X~U(0,1) and independent [duplicate]

I must have missed something important in the formulation of the problem. Can you please help clarify how should the following simple problem be formulated and where my mistake is. Let $Z = \frac{Y}{X}...
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1answer
57 views

What's the expression for convolution of a uniform[a,b] density and a normal(0,d^2) density?

Suppose I have $X\sim Uniform[a,b]$ and $Y\sim normal(0,d^2)$, what's the expression for the density of $Z=X+Y$? Let $F_{Z}(z)$ be the cdf of $Z$ evaluated at $z$, and let $\Phi(\cdot)$ and $\phi$ be ...
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21 views

How to test if numerical function describes a valid probability distribution?

Suppose I can query a function $f$, but I don't have its closed form. We know the following things about $f$: $f(x) \geq 0$ for all $x$ $f$ is continuous Additionally, I can choose whether $f(x) \leq ...
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1answer
24 views

Mixed Joint Probability

In the wikipedia definition they give the example of a logistic regression problem to predict $P(Y=y\,|\,X=x)$ where $Y$ is binary (discrete) and $X$ is a continuous random variable. Then the mixed ...
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18 views

Approximate ways of sampling from a joint distribution with multiple dimensions outside of trying to build a kernel density estimator?

I am trying to sample from a multivariate distribution of hourly data (e.g., imagine price or temperature values for every hour of the day). I want to generate scenarios of price/temperature data that ...
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76 views

Bounded probability distribution wanted

[Edit: this is a reformulation of the original question, based on the comments by whuber and LmnICE below] I am looking for a bounded continuous probability density function that occupies a finite ...
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143 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 ...
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26 views

which densities are Lipschitz in this sense?

Suppose $f(\cdot)$ is the pdf of some random variable with support $S\subset R$ ($S$ could equal to $R$). Suppose $x,y$ are two points in $S$, and $\beta_1,\beta_2$ are two scalars that belong to a ...
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If I want to model a bivariate distribution that is symmetric about (0,0) using copula, what copulas can I use?

If I want to model bivariate data $\{X_i,Y_i\}_{i=1}^{n}$ using copula. The true joint density of $(X,Y)$ denoted as $f_{XY}(,)$ is unknown, but I know it's symmetric about (0,0) in the sense that $f_{...
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252 views

If I know the density I'm estimating is symmetric about 0, how to impose this restriction in my kernel density estimator?

Suppose I'm interested in estimating the unknown smooth density of $X$ denoted by $f(\cdot)$ using data $\{X_i\}_{i=1}^{n}$. Suppose I also know that $f(\cdot)$ is symmetric about 0 in the sense that $...
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Are all these double integrals of the probability distribution of two continuous random variables equal?

If $X$ and $Y$ are two continuous random variables and $A$ and $B$ are any set within the range of $X$ and $Y$ respectively, are all these equal?: \begin{align*} P(X \in A, Y \in B) &=\int_{X \in ...
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PDF of squared random variables

i have complex gaussian random variable given as $h\sim\mathcal{C}\mathcal{N}(0,\sigma^{2})$. And i had $Y = |h|^{2}$. So what should be pdf of $Y$. I understood that $Y$ is exponential random ...
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25 views

Does minimizing KL-divergence result in maximum entropy principle?

The Kullback-Leibler divergence (or relative entropy) is a measure of how a probability distribution differs from another reference probability distribution. I want to know what connection it has to ...
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Is limiting density of discrete points (LDDP) equivalent to negative KL-divergence?

Is limiting density of discrete points (LDDP), which is a corrected version of differential entropy, equivalent to the negative KL-divergence (or relative entropy) between a density function $m(x)$ ...
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1answer
42 views

What does i.i.d. mean for multivariate case?

When we say a random variable is i.i.d., it's often used to describe the dependency between the observations of that random variable, which I call the row dimension, indexed by time if it's a time ...
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1answer
30 views

Is The Jacobian Needed to Find CDF for R in Polar Coordinates?

I'm attempting to use inversion sampling to generate points on a disk according to the following PDF: $$ f(r) = \dfrac{2}{\pi(1+r^2)} $$ Here, the polar angle would just be a uniform random variable ...
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69 views

What does this Statistic mean? And how to find a density of a statistic? [duplicate]

My First Question! But it's in two parts. Context: I am given a Probability Density Function, and the question wants me to find the density of a statistic. Given pdf: $$f(x, \theta, \phi)=\frac{1}{\...
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27 views

Probability Density Function and Maximum Likelihood Estimation for Multinomial Logistic Regression and GMM

I have some confusion about a few very basic concepts and terminology. Let's assume we have two models for classification, a multinomial logistic regression (MLR) model and a GMM classifier. I'm not ...
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28 views

Upper and lower limit of PDF

I have this example of a probability density function that is centered around 0. I would like to find out what are the lower and the upper limit of 95% of the data that is around 0. The goal is to ...
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19 views

Is the maximum entropy probability distribution only determined through comparison?

The maximum entropy probability distribution has entropy at least as great as that of all other members of a specified class of probability distributions (pdf's). Does that mean that the pdf with ...
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2answers
58 views

Do all random variables' probability distributions have entropy?

Entropy of probability distributions is the weighted average of the log probabilities of each observation of a random variable. Does this mean that every random variable that has a probability ...
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29 views

Confidence intervals for mixture of Gaussian distributions

I have a mixture distribution of 2 Gaussians. Here, the left has a weight of 0.1 and the right has a weight of 0.9. In this example, they have identical $\sigma$, but that may not always be the case....
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What's the $n-$ dimensional generalization of this formula?

Suppose random variable $X$ has a density $f(\cdot)$ that is symmetric about zero in the sense that $f(-x)=f(x)$, then we know that its cdf satisfies $F(x)=1-F(-x)$. My question is: what is the ...
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31 views

How to smooth an existing PDF?

I've generated a PDF of binned data using the python package binsmooth. The PDF is plotted in the following image: I am trying to smooth the PDF so as to provide a more intuitive interpretation of ...
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How to define and plot a distribution function in python?

I want to define a distribution function (gaussian or skewed,...), the X axis is from 0 to 255. I have the mode which is located at the point 100 and i have two points (40, 170) that i consider ...
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1answer
218 views

How to define a skewed normal distribution using mode and two points? [closed]

I want to define a Gaussian distribution function and plot it in python using the mode and inflection points parameter values instead of using the mean and standard deviation. For example, I have <...
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1answer
26 views

Finding C for a PMF of a frequency distribution

N has probability mass function: $p_o = p_1 =0$ and $p_k = c/k!$ for $k=2,3,4,...$ I used exp series $\sum_{n=1}^{\infty} \frac{x^k}{k!} = e^x$ to get $ c\sum_{n=1}^{\infty} \frac{1}{k!}$ then $ce=1$ ...
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1answer
19 views

Probability density function after transformation

Let $X,Z$ be random variables with probability density functions $p_X,p_Z$. Suppose $Z=f(X)$, where $f$ is continuous and differentiable. How is $p_Z$ related to $p_X$? It's tempting to say $p_Z(z) ...
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39 views

Obtaining marginal PDFs through change of variables

Given a random variable $\textbf{x} = (x_1, x_2, \ldots, x_D)$ with multiple dimensions and PDF $p_X(\textbf{x})$ and some invertible transformation $\textbf{y} = f(\textbf{x}) = (y_1, y_2, \ldots, ...
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1answer
77 views

Which financial time series have a PDF and/or a CDF? [closed]

Consider the following types of financial time series for a single publicly-listed stock: Price data Log returns Cumulative returns Each is computed from the item listed before it: log returns are ...
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1answer
45 views

Why copula based on CDF instead of PDF

I do understand the mathematic behind probability density function( PDF) and cumulative distribution function (CDF). My problem starts when I try to understand why copula relies on CDF and not on PDF. ...
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dbeta vs dbinom output comparison [duplicate]

My understanding of how density functions work in R is that they are calculations of the absolute probability (continuous) or probability mass (discrete) of something occurring, instead of the ...
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Another 3-part question, this time on limiting distributions. Care to critique my work?

Let $X \stackrel{d}{\sim} Geometric(p)$ for $0 < p < 1$. E.g., $X$ has the pmf $f(x|p) = p(1-p)^{x-1}, x = 1, 2, ...$ with $E(X) = \frac{1}{p}$ and $Var(X) = \frac{1-p}{p^2}.$ a.) Find the limit ...
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1answer
28 views

Interpretation of a mixture model

I want to have a mixture model like $\lambda \cdot P(s'|s, a) + (1- \lambda) \cdot P'(s'|s)$ where $P$ and $P'$ are conditional distributions and $\lambda \in [0,1]$ is a weight. I have two questions ...
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1answer
95 views

Change of variables in pdf

I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(...
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23 views

How can I interpret a kernel density plot where x axis is the probability?

I am confused on how to interpret the kernel density estimation (kde) plot given below, which is of the predicted probabilities from my model for class 0 and class 1. What conclusions can I draw from ...
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20 views

distribution of data vs density estimation

I think this is a very basic question but I'm confused and need clarification please help me density estimation vs distribution of data what is the difference and Is Kernel density estimation learns ...
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58 views

Finding PDF of max of sum of truncated sinusoids

Suppose I am adding N sinusoids. The frequencies, phases, and amplitudes are chosen to be iid in the range f1 to f2, 0 to 2pi, and A1 to A2 respectively. Now, if I am observing the sum from time t1 to ...
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34 views

Find $x_0$ that satifies $\mathbb{P}(X \leq x_0) = 0.75$

Suppose that $X$ is a continuous random variable with probability density function: $$\begin{cases} x & 0 \leq x < 1, \\[6pt] 2-x & 1 \leq x < 2, \\[6pt] 0 & \text{otherwise}. \\...
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23 views

Simulating p-value statistics for Liliefors test (Python statsmodels)

Python statsmodels has an implementation of Lilliefors' test for goodness of fit (i.e. if the parameters of the distribution were obtained from fitting the data and not per-determined as in the ...
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1answer
31 views

How to Evaluate a Probability Density Function you Fitted?

I was reading "A Student's Guide to Bayesian Statistics" by Ben Lambert and he brought up something I never thought of before and can't find the answer to on Google. That is, evaluating a ...
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15 views

How to estimate parameters and pdf of a random variable transformed from a lognormal random variable?

I have a continuous random variable Y that follows lognormal distribution with known parameters (mu and sigma). Let Y be transformed to X=Y-20000. So it is basically shifted to left. How do I find the ...
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20 views

Finding PMF, CDF of a piecewise function of an RV

Here's the question: Let $Z$ have CDF $F$ and pdf $f$ and let $A$ be a subset of the real line. Further, let \begin{cases} W = 1 & \text{if $Z \in A$} \\ ...
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1answer
42 views

3-part question on joint PDFs

a.) Let U, V be uniformly distributed over the set $\{(u,v): $$0<u<v<1$}. Let $X$ = $-$$log(U)$, $Y$ = $-$$log(V)$, $Z$ = $max$($X$,$Y$). a.) Draw the support of the joint distribution ($U$, $...
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1answer
53 views

Is it possible for a distribution to have a non-zero probability of generating a value with zero probability density?

I am reading the book "An Elementary Introduction to Statistical Learning Theory" and there is a sketch of a proof (Section 8.4) for the universal consistency of kernel rules for binary ...

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