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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7 views

What is the PDF for the minimum difference between a random number and a set of random numbers

I have a list (lets call it $ \{L_N\} $) of N random numbers $R\in(0,1)$ (chosen from a uniform distribution). Next, I roll another random number from the same distribution (let's call this number "b")...
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19 views

Sampling posterior distribution of a function

I have the following problem: let's say I have a function $y=f(x)$. Let $f$ be defined for all $x$ but it it might not be invertible. Further assume $x \sim p(x)$ with some probability density $p(x)$. ...
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Why is the probability mass function of a transformed discrete random variable summed over the inverse values of the function?

Let $X$ be a random discrete variable with probability mass function (pmf) of $p_X(x) = P(X = x)$. Let $Y = g(X)$ (from $\mathbb{R}$ to $\mathbb{R}$). Then, why is it that: $$p_Y(y) = \sum_{x \in g^{-...
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Questions about pmf of multinomial distribution with indicator variables

I was reading through the textbook of Introduction to Machine Learning, it introduces $z^{t} = (z_{1}^{t},...,z_{k}^{t})$, where $z_{i}$ is an indicator variable, each with probability $\pi_{i}$, then ...
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33 views

Simplifying modified Bessel function of the first kind

The modified Bessel function of the first kind shows up in the normalizing constant of a lot of random variables (e.g. the normal product distribution, the noncentral chi-square distribution, the ...
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15 views

Does the below Plot refer to a Valid PDF?

I have tried to creat a valid PDF according to some special function I have got the below Plot , I got it for n=6 Integrand Range[n/n], Now My question here Is : Is that below Plot refer to a valid ...
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GANs: How do I interpret the pdf of the sample data - p(x)?

In the context of GANs, the concept of a probability distribution comes up as the generator tries to emulate the "distribution" of the data: $p_{data}(x)$. For me, the use of "distribution" here ...
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A Multinomial Geometric distribution? What is this distribution called?

In a problem I am dealing with I repeatedly interact with a distribution of the following form, $$p(n_1, n_2, \ldots n_M)=\binom{\sum_{i=1}^Mn_i}{n_1\cdots n_m} w_0\prod_{i=1}^Mw_i^{n_i}$$ where $p$ ...
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21 views

problem in understand exponential PDF [duplicate]

I'm studying a paper called "Optimization based on bacterial chemotaxis". As it can be understood from its name, it has proposed an optimization algorithm based on the reaction of a bacterium toward ...
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Calculating the truncated version of the squared hyperbolic secant PDF

$ \newcommand{\sech}{\mathop{\rm sech}\nolimits} $ Hello, I have the following Probability Density Function (PDF): $f(x)=\frac{1}{2s}(\sech\frac{x}{s})^2$ This PDF has support for $x\in(-\...
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Spectral Density of a Stationary, Isotropic Gaussian Kernel

I intend to perform a simulation of a Gaussian process. To that end, I use a stationary, isotropic Gaussian covariance function (aka Gaussian kernel, or squared exponential kernel), $k(r)= \exp (-\...
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38 views

Finding distribution of $\frac{X^2}{Y}$ given joint pdf of $X$ and $Y$

So I have the joint distribution of $X$ and $Y$: \begin{align} f(x,y) = cx, \ 0 < x^2 < y < \sqrt{x} < 1 \end{align} and I want to find the distribution of $X^2/Y$. So I set $U = X^2/Y$ ...
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50 views

Why density plot tails are beyond maximum and minimum values?

I am trying to interpret the tails of a density curve, which go beyond xlims(0 in this case). I understand that area under the curve between any two points represents the probability of that event. ...
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35 views

How to calculate the PDF of the 'difference' between two Beta distributions?

I start with two Beta distributions: $$\mathrm{Beta_A}(p; \alpha_A, \beta_A) = \frac{p^{\alpha_A-1}\,(1-p)^{\beta_A-1}}{\mathrm{B}(\alpha_A, \beta_A)}$$ $$\mathrm{Beta_B}(p; \alpha_B, \beta_B) = \...
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Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?

What the title says! My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
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27 views

How do you calculate the expectation without a pdf in the context of Center limit theorem, variance

Given Problem 1) To get the variance and covariance the following steps are taken: In the step below to calculate E[Z^2] how do we approach this without a known pdf? For completeness would finding E[...
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27 views

Is this a valid pdf

$f_x (x)= x$ if $x \in [0,1]$ and $f_x = 0$ otherwise. Is this a valid pdf? It seems to me it is not since the area under the pdf is 0.5.
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Calculation of joint PDF

we have the joint PDF of two RVs $X$ and $Y.$ we also have two RVs $U = f(X,Y)$ and $V = g(X,Y),$ where $f$ and $g$ are two variable functions. How can I calculate the joint PDF of $U$ and $V$? for ...
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25 views

Conditional and density probability (normal distribution)

I am trying to solve the following problem: Suppose that $\mu\sim N(1,4)$ and $Y|\mu\sim N(\mu,1)$. Show that: $$\begin{bmatrix}Y \\ \mu \end{bmatrix} \sim N\bigg(\begin{bmatrix}1 \\ 1 \end{bmatrix},...
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How can I calculate this PDF

I'm trying to reproduce the results of a ray tracing paper which uses reinforcement learning. I asked my question in the computer graphics community of this site, but I think my problem can easily be ...
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37 views

Sum of two continuous random variables

Let R1 and R2 be two independent random variables, both with uniform density at the interval (0,2). What is the probability of R1>1 given that R1 +R2<2? -- What I've tried: I know that $$ P(R1&...
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43 views

Find conditional pdf given joint

Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$. What is the marginal cdf of $X$? of $Y$? Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
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21 views

getting a density function from a sample of 500

I'm pretty new to statistics so please excuse me if the answer is obvious. The scenario is the following: I am using mcmc to sample from a posterior distribution of a parameter. I then need to ...
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66 views

Mixing probabilities and probability densities

I'm currently working on a Bayesian network designed to find the probabilities for various lung diseases. In the network there are, among others, a normally distributed random variable (body ...
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Gaussian Distribution [duplicate]

Assume we have two continuous Normal RV "X" and "Y". how can I show the conditional PDF f(X|Y) and f(Y|X) is Normal?
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70 views

Given a pmf, how is it possible to calculate the cdf?

Given a pmf (probability mass function) for X (random variable): \begin{array}{|c|c|c|c|c|}\hline x&1&2&3&4\\ \hline p(x)&0.4&0.3&0.2&0.1\\ \hline \end{array} How ...
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How to obtain value range of empirical PDF, given the mode and area?

Given an empirical PDF of a continuous random variable $X$, then integrating over its entire defined domain will yield an area of size 1. To find the probability of $X \ge x_1 \land X \le x_2 $ (as in ...
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23 views

how to find domain of marginal pdf when its two variables domain are dependent

I have a pdf $f(x,y)=1/π, 0< x^2+ y^2 <1$; 0, e.w. Here, we can see $-\sqrt{1-x^2} < y < \sqrt{1-x^2}$ So, the marginal pdf of $X$ is $$\int_{-\sqrt{1-x^2}}^\sqrt{1-x^2} 1/πy \, dy\,.$$ ...
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45 views

Example: Writing the joint PDF $f(x, y)$ as the product of a marginal and a conditional probability function

I am presented with the following notes on Bivariate distribtions: If we can write the joint probability density function $f(x, y)$ of a pair of random variables $(X, Y)$ as the product of a ...
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49 views

Behavior of kernel density estimation

Consider the random variable $X=YZ$, where $Y\sim\text{Normal}(0,1)$ and $Z\sim\text{log-Normal}(0,1)$ are independent. I wanted to assess the accuracy of kernel density estimates for the density ...
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80 views

Acceptance/rejection sampling and inverting CDF (R code illustration included)

I have the following example: Acceptance/rejection sampling In some cases the cumulative distribution function might not be (easily) invertible. For example if $X$ has the probability density ...
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38 views

What is the difference between Empirical and analytical PDF and CDF? More Precisely what would be the difference in their plotting?

I am relatively new to statistics with no statistical background whatsoever, I have an assignment in which i have to plot different distributions in these four manners, i have a gist of empirical PDF ...
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Mixture Densities weights

I'm supposed to find the mixture weights and densities of all the mixture components. Should i find the normalizing constant in this case then work from there? Any hints or solutions will be much ...
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38 views

quickly finding the moments of a numerically-defined PDF

I have a two-dimensional continuous PDF which is numerically defined. From this PDF I would like to extract the second central moment (variance) of a "slice" of this distribution. The "slice" is to ...
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16 views

Dynamically updating posterior density in R

I want to redefine my function in a loop by calling the function from last iteration. However I know this is basically a recursive way which I don't want. To give an example, see the following ...
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34 views

Find $\alpha$ and $\beta$ so that $f_X(x)$ can be a density function

\begin{equation*} f_X(x) = \begin{cases} \frac{4x^2}{5} & \text{ , if } 0 < x \leq 1\\ \alpha(5-2x) & \text{ , if } 1 \leq x < 2\\ \beta x^2 & \text{...
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Density estimation as an optimization problem

Density estimation is the estimation of a probability density function from observed data. Can some of the common approaches to density estimation, such as kernel density estimation, be formulated as ...
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14 views

Calculating a baseline probability model for images

I'm a newbie to statistics, so I apologize if this question is trivial. I'm trying to build a distribution that can predict a specific set of images. But first, I need a baseline - so, I decided to ...
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Bivariate Dist Study Question Help - determine joint PMF and P( … )

I am in a prob. models class. Current module is on Bivariate and Multivariate Distributions. The question below has me stumped though. It is from a study guide and I would like to know the answer ...
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What is the name of this distribution?

I came across this: a categorical distribution with $K=10,000$ parameters (categories), and we take only few samples from this distribution, say $N=400$ (the point is $N < K$). Now, obviously, not ...
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66 views

how to scale the density plot for my histogram

I have the histogram plot and I'd like to overlap it with density line for the same data. Importantly, I don't want to turn histogram into density values, but want to keep N (numbers) on y axis. Is ...
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20 views

What is the distribution of a point to a random line?

Let's say that we have a random vector that represents a line [A,B,C] so that, Ax + By + c = 0. A, B, and C are independent Normally distributed random variables, with different mean and variances. if ...
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29 views

How do we find the constants of pdf using the $\mathrm{E}(X)$? [closed]

I have a pdf of cont random variable as $(a+bx^2)$ between $0$ and $1$. how do I find the constants $a$ and $b$ when my $\mathrm{E}(X)=3/5$?
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What is the probability distribution used in logistic regression called?

In logistic regression, we set the probability of predicting a target $y$ given a data $x$ as, $\Pr(Y = 1|X;w) = \dfrac{\exp(w^TX)}{(1+\exp(w^TX))}$ What is exactly this probability distribution (or ...
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73 views

How to calculate the probability that the median exceeds a certain value?

Given the pdf $f(x)=\begin{cases}2x&\text{0<x<1}\\0 & \text{otherwise}\end{cases}$. What is the probability that the sample median based on a random sample of size 3 drawn from the ...
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1answer
35 views

Strobiloid representation

I am working with household income and I would like to replicate the following strobiloids generated by Chauvel (2013): For what I understand here and from his explanation of the graph, he plots the ...
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Choice probabilities and integration

I have a question from the book "Discrete Choice Methods with Simulation" (Train, 2009). In chapter 1, , part 1.2.3, partial simulation, partial closed form, it has the following equation which ...
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52 views

How to fully estimate a probability density from only a sample of minimum values?

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., $z = \min \{ x_1, x_2,\ldots,x_n \}$. By means of ...
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27 views

Uniqueness of change of variable function

Let $X$ and $Y$ be continuous random variables with probability density function as $p_x(X)$ and $p_y(Y)$. If $X$ and $Y$ are related by an invertible function $f$ as $f(X)=Y$, then using change of ...
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67 views

What's the big deal with normalization constants in Bayesian inference? [duplicate]

I read this sentence in a book: "... therefore this method is particularly useful for Bayesian inference since it doesn't require a normalization constant" The method is a computational algorithm ...