# Questions tagged [density-function]

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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### Distribution/expected length of the shortest path in infinite random geometric graphs

Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
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### What does it mean for a probability distribution to not have a density function?

I understand the distinction between probability mass and density functions. But I don't understand what it means for a continuous random variable to have a probability distribution but not a density. ...
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### Spinograms vs. conditional densityplots

I have a binary response variable (hail) and multiple continuous predictor variables. My aim is to understand the linear/non-linear relationship of the predictors to the response to be able to justify ...
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### 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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### Expectation of a strictly increasing function

Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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### Testing for Normality (CDF)

I was reading an article about using the CDF to calculate the area between 2 points on the normal curve. They gave a sample of 7 for illustration purposes: ...
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### Problem involving P.D.F. containing an indicator variable

Let $X_1, X_2, \ldots$ be independently and identically distributed random variables with probability density functions: $$f(x) = \alpha \;x^{-(\alpha+1)} \; I_{(x>1)}, \; \; \alpha > 0.$$ For ...
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### CDF of the ratio of two correlated $\chi^2$ random variables

It is well known that the sum of a series $m$ of squared standard independent normal random variables follows a $\chi^2$ dstribution with $m$ degrees of freedom. It is also true that the ratio of two ...
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### Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
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### Geometric construction of copula - question regarding C-volume

I am learning about copula's, using Nelsen's book, and more specifically about the geometric method of constructing copula's. The problem is replicated in the following link: http://www.stat.ubc.ca/...
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### Maximum likelihood estimation involving both probabilities and probability densities

Note: based on suggestions in the comments, I have rewritten this question. Please refer to the history for the original version. In general my question regards how to compute likelihoods in mixed ...
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### Expectation of density ratio of two iid variables

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio \begin{align} \mathbb{E}\left[\frac{f(X)}{f(Y)...
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### Confusion related to Parzen window

I was going through this tutorial related to Parzen window at http://www.cs.utah.edu/~suyash/Dissertation_html/node11.html. However, I have some confusion related to Parzen window with gaussian kernel ...
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### Density function of a dependent sum of products of normal random variables

Say we have a random variable $$X = A_0 A_1 + A_0 A_2 + A_1 A_2,$$ which consists of normally distributed independent random variables $A_0, A_1, A_2 \sim \mathcal{N}(0,1)$ with probability ...
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### What is p(data) in image generation

In the context of image generation architectures such as VAEs or GANs (say we are using mnist digits) what do we mean by probability distribution of the data? Just to clarify this question and make it ...
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### Integral of difference of density functions of two Continuous Random Variables goes to 0

The problem says : Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
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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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### 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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### Fitting Pareto distribution to data example in SciPy

In docs.scipy.org there's code to sample data from a Pareto distribution and then fit a curve on top of the sampled data. I could understand most of the code snippet except the term ...
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### What methods are there for estimating distributions based on histograms?

I recently worked on a consulting project where a client wanted to estimate gamma and weibull distributions based purely on histograms rather than raw-data. I have never worked with problems like that ...
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### Calculating a Confidence Interval for a Proportion for a Sample of Different Size

I'm interested in a (preferably analytic) solution or approximation to the following problem: Let $s_1$ be a sample from an unknown distribution of size $N_1$ and with proportion of successes $p_1$. ...
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### Sampling from $f(x)$ given approximation $g(x)$

(After some pondering, what I really wanted to ask is how to incorporate prior information about $f$ into a sampling method - see this question.) Suppose you want to draw samples from an (...
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### Probability Density Function of a linear combination of 2 dependent random variables, when joint density is known

Let's say there are two dependent random variables $X$ and $Y$ with joint density function $f$. What is the PDF of the weighted sum of these two variables, $Z = aX + bY$? Thanks in advance for any ...
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### Error bounds when approximating densities

I was curious whether it is possible to obtain approximation error bounds on continuous densities from approximation error bounds of random variables. To make my question more precise: We consider ...
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