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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22 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
24 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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1answer
64 views

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

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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18 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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25 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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13 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
48 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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16 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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18 views

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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1answer
28 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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33 views

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
46 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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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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36 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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Which financial time series have a PDF and/or a CDF?

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
35 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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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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86 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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Differentiation of Erlang Cumulative Distribution Function (CDF) analytically [migrated]

I have an idea of Erlang PDF by intuition but I want to get the answer by analytical derivation of its CDF i.e. \begin{equation} F_{Y_k}(y)=1-\sum_{n=0}^{k-1}\dfrac{(\lambda y)^n e^{-\lambda y}}{n!} \...
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18 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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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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56 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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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
28 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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11 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
52 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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104 views

Does this distribution have a name? $p(x) \propto |x|^a \exp\left(-\frac{1}{2} (x-b)^2 \right)$

Quick question. Anyone able to attribute the following kernel to a known probability distribution (univariate, continous on the real line)? $$ p(x) \propto |x|^a \exp\left(-\frac{1}{2} (x-b)^2 \...
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Expectation (x/y) using jacobian

Let $X$ and $Y$ be two independent random variables with the density functions: $f(x) = 3 x^2$, for $0<x<1$, $0$ elsewhere $g(y) = 4y^3$, for $0 <y<1$, $0$ elsewhere Give $\mathbb E(x/...
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1answer
57 views

finding PDF of Y, given Y|X [closed]

$$Y|X\sim Bin(X,n)$$ $$X\sim U([0,1])$$ How can I find the PDF of Y? I know that: $$\Bbb P(Y=k)=E_X[\Bbb P(Y=k)|X]$$
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How can we perform the integration for showing this equation?

Im reading this paper momentarily, and there is one equation (9) in section 3.1. that I just can't wrap my head around yet: \begin{align} \mathcal{N}(\textbf{y}_d;\textbf{0},\pmb{\Phi \Phi}^T + \...
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1answer
21 views

Calculate the likelihood from the density function with known mean and sd

In this link the likelihood of IQ has been calculated by using dnorm function in R. Here they used "%" sign but based on the range of ...
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2answers
44 views

How do I find the PDF from a multidimensional CDF with indicator functions?

I have what I'm sure is a very stupid question. When I have a two-dimensional random variable $\tilde{X}=(X_1,X_2)$ with the cdf $F(x_1,x_2)=(kx_1^2I_{(0,1)}(x_1)+I_{[1,\infty)}(x_1))(kx_2^2I_{(0,1)}(...
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What are some examples of multiparameter probability distributions with three or more parameters?

What are some examples of multiparameter probability distributions with three or more parameters? What real life phenomena are they used for modelling?
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37 views

How do I show this equation involving the Gaussian pdf? [duplicate]

Im reading this paper momentarily, and there is one equation (9) in section 3.1. that I just can't wrap my head around yet: \begin{align} \mathcal{N}(\textbf{y}_d;\textbf{0},\pmb{\Phi \Phi}^T + \...
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1answer
37 views

Density of square root of sum of squared independent uniform random variables [duplicate]

Let $X \sim U(-1, 1)$ and $X \sim U(-1,1)$. We want to find density function of $W = \sqrt{X^2 + Y^2}$. I got stuck and I have no idea, where I am making a mistake. This is my approach. Let $F$ be a ...
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1answer
79 views

Conditional probability density from probabilities

I am trying to understand conditional probablility densities in relation to the conditional probablilities. From the Measure-theoretic definition on Wikipedia, if $X$ and $Y$ are non-degenerate and ...
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1answer
43 views

How to get a PDF which converts an already drawn sample to uniform [closed]

Suppose i have a large data pool with a particular PDF, $F(x)$, interval $[x,y]$ estimated from KDE of the datapool. I drew $N$ samples at random from that data pool and saw that their distribution is ...
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1answer
36 views

Marginal distribution

A loss distribution has PDF - $f(x) = 1/x^2$, for $x > 1$ An insurer finds that the time in hours it takes to process a loss amount x has a uniform distribution on the interval $(\sqrt x, 2\sqrt x)$...
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1answer
45 views

An example of continuous random variable X > 0 with finite second moment but Infinite third moment [duplicate]

Can someone construct an example of this? i.e., $E[X^2] < \infty$ but $E[X^3] = \infty$. Results could be in terms of pdf, or cdf, or survival function. Justification would be appreciated
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1answer
25 views

Finding P(a< u(X,Y) <b) given a rectangular support

>The continuous variables X and Y have the following joint pdf $f(x,y) = x + y, 0<x,y<1.$ Determine $P(0.5<X+Y<1.5)$. I know that the support of x and y is rectangular, hence they are ...
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1answer
33 views

How do you work with a function of a uniform distribution? [closed]

I am struggling with parts b and c. How do you solve them? Could you please give the solution?
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14 views

evaluating an empirical multivariate PDF in python

I have multivariate (bivariate in the simplest case) residuals from a VAR time series regression and I'd like to estimate the joint pdf and then be able to draw from this pdf. If I have bivariate ...
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1answer
24 views

Is pX(Y) a random variable or a number?

I reason that is a random variable because Y is a random variable, thus making Px acting randomly. Example Y sample space is a roll of a die (1,2,3,4,5,6). So any of those values could be inputed in ...
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1answer
27 views

Why does Uniform distribution make sense?

This might be a dumb question, but I am suddenly confused on how to understand the PDF of a uniform distribution. For instance, the PDF of standard uniform is always equal to 1... How is that ...

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