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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1answer
30 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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1answer
124 views

Combine Multiple Discrete Probability Density Functions

I'm a bit stuck trying to figure out the combined probability from several discrete PDF's. Lets say I have a bunch of different classes (Truck, Sports Car, Station Wagon, etc) and a bunch of ...
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Practical applications of the Laplace and Cauchy distributions

I want to know if there are any examples of real-life applications of the Laplace and Cauchy density functions. How do they differ in their applications? This related post, however, does not answer ...
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1answer
25 views

Finding a distribution of a transformation of a random variable [on hold]

Suppose X ∼ Exponential(λ). Find the distribution of the random variable Y = λX, where λ > 0 So do I just multiply the exponential distribution by another lambda?
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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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2answers
80 views

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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1answer
297 views

Difference between probability density functions and sampling distributions

I was wondering what is/are the fundamental difference(s) between a probability density function for a mean and sampling distribution of a mean? Can we say that ...
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1answer
203 views

Parametric Definition of Skewed Normal Distribution with Left and Right Percentile

Is it possible to easily build a Skewed Normal Distribution with these 3 parameters? -Mean (or median) 99.7-th Percentile for data to the left of the mean (median) 99.7-th Percentile for data to the ...
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1answer
26 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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1answer
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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24 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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48 views

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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176 views

Estimating an underlying pdf from binomial trials

I'm afraid I'm not an expert in statistics, but I have a particular problem I'm interested in solving. I'm pretty sure this area already has a lot of literature, but I'm having difficulty finding ...
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1answer
127 views

Excepted conditional density and conditional expectation

Apparently one can obtain a regression analysis as $$g(x)=\frac{\int yf(y,x)dy}{f(x)}$$ where $$f(x)=\int f(y,x)dy$$ is the marginal density of $X_i$. In effect, I believe, the above expression ...
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35k views

How to find the mode of a probability density function?

Inspired by my other question, I would like to ask how does one find the mode of a probability density function (PDF) of a function $f(x)$? Is there any "cook-book" procedure for this? Apparently, ...
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1answer
36 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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2answers
92 views

Mode estimation in high dimensions

Suppose we have a sample $\boldsymbol{x}_i$ for $i$ in $1,\dots, n$, from a $d$-dimensional unimodal density $f(\boldsymbol{x})$. I would like to estimate the mode of $f(\boldsymbol{x})$. The mean-...
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1answer
142 views

Fast multivariate unimodal density estimator

I have a sample $\boldsymbol{x}_i$ for $i$ in $1,\dots, n$, from a $d$ dimensional density $f(\boldsymbol{x})$ and I would like to estimate this unknown density. In addition I know that $f(\boldsymbol{...
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1answer
42 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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1answer
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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2answers
59 views

Mixing probabilities and probability densities

I'm new to the field so I apologize in advance if this is a stupid question. I'm currently working on a Bayesian network designed to find the probabilities for various lung diseases. In the network ...
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17 views

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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2answers
63 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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10answers
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Why is the sum of two random variables a convolution?

For long time I did not understand why the "sum" of two random variables is their convolution, whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum ...
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0answers
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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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1answer
21 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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2answers
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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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2answers
458 views

Does Fisher's factorization theorem provide the pdf of the sufficient statistic?

From Wikipedia Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is $ƒ_θ(x)$, then $T$ ...
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1answer
44 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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43 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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1answer
36 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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0answers
6 views

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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37 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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3answers
320 views

Why is the Hazard function not a pdf?

I am trying to understand why the hazard function is not a PDF. For a random variable T, people often define the PDF of this random variable as: $$f(t)=\lim_{\delta \to 0} \frac{P(t\leqslant T <t+...
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4answers
19k views

“The total area underneath a probability density function is 1” - relative to what?

Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%. But I ...
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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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1answer
57 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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12 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
23 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
47 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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1answer
60 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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48 views

Simulating from an Epanechnikov kernel density estimate in MATLAB / exact form of the Epanechnikov kernel in MATLAB?

It's my first time posting, so apologies if I'm breaking any etiquette. I've used MATLAB's ksdensity function to estimate a density using the Epanechnikov kernel and would now like to make repeated ...
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1answer
166 views

Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. Note that the $X_i$ are independent, but not identically distributed. That is, ...
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3answers
856 views

Whence the beta distribution?

As I'm sure everyone here knows already, the PDF of the Beta distribution $X \sim B(a,b)$ is given by $f(x) = \frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1}$ I've been hunting all over the place for an ...
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0answers
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
1k views

Marginal distribution of the diagonal of an inverse Wishart distributed matrix

Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the marginal distribution of the diagonal elements $\operatorname{diag}(X) = (x_{11}, \dots, x_{pp})$. There are a few ...
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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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2answers
98 views

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 ...