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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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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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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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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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Plotting a PDF of Exponential random Variable in python [on hold]

I want to plot a pdf of an Exponential random variable that i have generated using: ExponentionalRV = np.random.exponential(scale=1.0, size=1000) How do i go ...
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1answer
35 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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27 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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15 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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32 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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1answer
51 views

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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10 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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1answer
21 views

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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1answer
56 views

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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1answer
38 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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1answer
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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2answers
70 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
30 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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17 views

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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50 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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21 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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62 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 ...
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14 views

How to fit mixture of gaussians with identical mean?

Say I have data generated by a mixture of gaussians whose components have the same mean, but very different covariances, like the one generated by this code: ...
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1answer
57 views

Probability mass function of product of two binomial variables

I have two i.i.d. binomial variables $X$ and $Y$ with given $n$ and $p.$ What is probability mass function of $Z = X \times Y$? I need pmf as function $f(Z, n, p).$
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What is the PDF sum of N random variables

I have N random variables: $X_1,...,X_N$ which are all independent. The PDF (probability density function) of each random variable $f_{X_i}=e^{-a/(x^{2/b})}x^{-(4+b)/b}$. What is the PDF $f_S(x)$ ...
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43 views

Is the difference between two samples of two distributions a sample of the difference of the distributions? [closed]

My question is as follows. Suppose you have two variables $X$, $Y$. If I pick a random sample $x$ from the distribution over $X$ that has probability density function $f_X$, and another random sample $...
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2answers
84 views

Differences between a frequentist and a Bayesian density prediction

What are some essential differences between a frequentist density forecast/prediction and a Bayesian posterior for an outcome of a random variable? Of course, there will be differences in how they ...
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32 views

How to find pdf of transformed r.v using joint distribution

I'm intrigued by the following idea but I don't know how to do it. If I have a r.v. $x$ with given distribution $f_X(x)$ and I have a second variable $y=2x$. The goal is to find $f_Y(y)$. I know the ...
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30 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 ...
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24 views

Condition for this function of conditional probability density to be increasing

Let $Y$ and $W$ be two jointly distributed random variables. The conditional probability density of $W$ given $Y$ is given by $f_{W|Y}$, assumed to be continuous and twice differentiable. Let $X$ ...
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Looking at two PDF plots, is it possible to guess which distribution has a greater Gini coefficient?

By observing the PDF of two different distributions over the same support (as in the image), is it possible to infer which PDF describes the distribution with the greater Gini coefficient? I assume ...
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3answers
127 views

M buckets out of N have at least one ball

We have N buckets and we start filling them randomly with balls. At the end we know that we have exactly M buckets that have at least one ball in them. Both N and M are given and M < N. What is ...
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1answer
71 views

Calculating the sum of dependent uniform random variables

My question derives from Problem calculating joint and marginal distribution of two uniform distributions. So, suppose we have random variables $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as ...
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14 views

Mix pdf and cdf in binary response model [duplicate]

Let's suppose that I have a model that tells me how likely is for an event to have come after a certain time lapsed, given by some kind an exponential distribution, i.e. $$ \mathbb{P}(T_E < t) = \...
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43 views

Kernel Density Estimation - Comparison Between different sets of samples

Is there a way for compare the distribution of different set of samples? For example, I have three sets, for example: X1 = N(0, 1); X2 = N(0.5, 1); X3 = N(1, 1). Each set is drown with a specific (...
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1answer
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Distribution of inbag matrix when sampling with replacement

Say I take a random sample of size $M$ from a sample of size $N$, like, for example you'd do when bootstrapping in random forest. As you increase $M$, you're more likely to sample any particular ...
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1answer
89 views

Find probability between $-\infty$ and $0$

The graph shown below is the numerically result of differences of Normal Distribution ($N(15.5 , 0.60^2))$ and Exponential Distribution $(\exp(0.5))$ (Both are independent). I am trying to find the ...
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1answer
113 views

Correct or Not? Probability and Geometry [closed]

A stick of length $1$ is broken into two pieces of length $Y$ and $1−Y$ respectively, where $Y$ is uniformly distributed on $[0,1]$. Let $R$ be the ratio of the length of the shorter to the ...
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1answer
38 views

CDF for f(x) = 0.5e^-|x|

This is the full question: "If a random variable has density f(x)= 0.5e^-|x|, for x∈R, find the cumulative distribution function". I know that to find cdf from the pdf you would have to integrate ...
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Does Wolfram Mathworld make a mistake describing a discrete probability distribution with a probability density function?

Usually a probability distribution over discrete variables is described using a probability mass function (PMF): When working with continuous random variables, we describe probability distributions ...
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1answer
56 views

What information does a probability density function (PDF) graph provide?

This sounds like a simple question and I know PDF graphs are used a lot in presentations and financial publications. Yet, what information does it actually provide? The CDF actually gives you ...
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1answer
66 views

Probability density function for continuous random variable

The question might be very basic and stupid on certain levels, but please help me out here!! I recently picked up stats and went through discrete and continuous random variables. Discrete variables ...
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0answers
67 views

Convexity of conditional expectation

Define $g(k)\equiv\mathbb{E}(X|_{X>k})$ and assume that the probability density $f$ of $X$ is twice continuously differentiable. Is there a sufficient condition in terms of $f$ that imply that $g^{...
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The dynamics of a normal distribuition in stochastic processes (food court example)

Suppose I want model a huge food court. Let's assume that the number of people who start having a lunch is a function of time $f(t)$. Also, let's consider that the time people spend having a lunch ...
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Sampling from joint distribution by writing its density as a product of conditional densities

In Gelman et al. "Bayesian Data Analysis Ed3" the authors often do the following (e.g. on pg. 65): Given two parameters $\mu$ and $\sigma^2$ and data y joint posterior density $p(\mu,\sigma^2)$ is ...
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Quality of a quantile regression learner

Given a learning algorithm that selects and trains quantile models, how do we evaluate it? One idea is to - use the algorithm to train a model on a synthetic dataset with labels drawn from an ...
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Formalism to cope with probability density functions defined piecewise (in the second dimension)

I'm not sure how to pose this question, as I lack the correct terminology. Actually, my question tries to obtain insight on the terminology and notation to cope with the following problem: I have a ...
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1answer
71 views

Computing probability density function at a point, given the covariance matrix and mean

(Edited for clarity.) Say I have the variance-covariance matrix $\mathbf{V}$ and mean $\mathbf{\mu}$ of a multivariate normal distribution. Given a sample, $\mathbf{s}$, can I compute/estimate the ...
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2answers
61 views

Intuitive explanation of “density generators”?

I was reading through Meucci's Risk and Asset Allocation (2005), when I happened upon the concept of a "density generator", which I have not been able to find good explanations for anywhere online, ...