# 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.

1,708 questions
Filter by
Sorted by
Tagged with
125k views

### Can a probability distribution value exceeding 1 be OK?

On the Wikipedia page about naive Bayes classifiers, there is this line: $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve ...
• 4,839
115k views

### Why does the Cauchy distribution have no mean?

From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
• 6,279
55k views

### What is the reason that a likelihood function is not a pdf?

What is the reason that a likelihood function is not a pdf (probability density function)?
• 1,576
185k views

### How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?

Given the random variable $$Y = \max(X_1, X_2, \ldots, X_n)$$ where $X_i$ are IID uniform variables, how do I calculate the PDF of $Y$?
• 861
74k views

### 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 ...
• 13.2k
8k views

### Are CDFs more fundamental than PDFs?

My stat prof basically said, if given one of the following three, you can find the other two: Cumulative distribution function Moment Generating Function Probability Density Function But my ...
• 4,373
10k views

• 1,244
2k 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")...
28k views

### How to get ellipse region from bivariate normal distributed data?

I have data which looks like: I tried to apply normal distribution (kernel density estimation works better, but I don't need such great precision) on it and it works quite well. Density plot makes a ...
• 315
3k views

### 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 ...
• 556
5k views

### Where is density estimation useful?

After going through some slightly terse mathematics, I think I have a slight intuition of kernel density estimation. But I am also aware that estimating multivariate density for more than three ...
• 469
21k views

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. My thought process so far: \begin{align*} F_Y(y) &= \mathbb{P}(\max(... • 301 16 votes 4 answers 15k views ### Area under the "pdf" in kernel density estimation in R I am trying to use the 'density' function in R to do kernel density estimates. I am having some difficulty interpreting the results and comparing various datasets as it seems the area under the curve ... • 2,402 16 votes 1 answer 46k views ### How to interpret height of density plot How should I interpret the height of density plots: For example in the above plot, peak is at about 0.07 at x=18. Can I infer that about 7% of values are around 18? Can I be more specific than that? ... • 10.1k 16 votes 1 answer 2k views ### Linear transformation of a random variable by a tall rectangular matrix Let's say we have a random vector \vec{X} \in \mathbb{R}^n, drawn from a distribution with probability density function f_\vec{X}(\vec{x}). If we linearly transform it by a full-rank n \times n ... • 321 15 votes 3 answers 8k views ### Closed form formula for distribution function including skewness and kurtosis? Is there such a formula? Given a set of data for which the mean, variance, skewness and kurtosis is known, or can be measured, is there a single formula which can be used to calculate the probability ... • 4,839 15 votes 4 answers 2k views ### Are the terms probability density function and probability distribution (or just "distribution") interchangeable? Like the title says, are the terms probability density function and probability distribution (or just "distribution") interchangeable? If not, what is the difference? • 151 15 votes 1 answer 12k views ### Is there an optimal bandwidth for a kernel density estimator of derivatives? I need to estimate the density function based on a set of observations using the kernel density estimator. Based on the same set of observations, I also need to estimate the first and second ... • 1,173 15 votes 1 answer 10k views ### Finding local extrema of a density function using splines I am trying to find the local maxima for a probability density function (found using R's density method). I cannot do a simple "look around neighbors" method (where ... • 373 15 votes 1 answer 412 views ### Are there non-trivial settings where the MAD statistic has a closed-form density? The MAD statistic of an iid sample (x_1,\ldots,x_n) is defined as the median of the absolute deviation from the median: \text{mad}(x_1,\ldots,x_n)=\text{med}\left\{|x_i-\text{med}(x_1,\ldots,x_n)|...
• 106k
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 ...
I keep seeing density functions that don't explicitly arise from conditioning written with the conditional sign: For example for the density of the Gaussian $N(\mu,\sigma)$ why write:  f(x| \mu, \...