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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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### 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 ...
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### 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?
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### 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)?
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### 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 $Y$?
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### 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 ...
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### Which to believe: Kolmogorov-Smirnov test or Q-Q plot?

I'm trying to determine if my dataset of continuous data follows a gamma distribution with parameters shape $=$ 1.7 and rate $=$ 0.000063. The problem is when I use R to create a Q-Q plot of my ...
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### Why does a Cumulative Distribution Function (CDF) uniquely define a distribution?

I have always been told a CDF is unique however a PDF/PMF is not unique, why is that ? Can you give an example where a PDF/PMF is not unique ?
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### 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 ...
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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 ...
34k 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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### How to find/estimate probability density function from density function in R

Suppose that I have a variable like X with unknown distribution. In Mathematica, by using SmoothKernelDensity function we can ...
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### How to calculate overlap between empirical probability densities?

I'm looking for a method to calculate the area of overlap between two kernel density estimates in R, as a measure of similarity between two samples. To clarify, in the following example, I would need ...
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### 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 ...
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### 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 ...
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### Best way to put two histograms on same scale?

Let's say I have two distributions I want to compare in detail, i.e. in a way that makes shape, scale and shift easily visible. One good way to do this is to plot a histogram for each distribution, ...
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According to the Wikipedia article on the Gamma distribution: If $X\sim\mathrm{Gamma}(a,\theta)$ and $Y\sim\mathrm{Gamma}(b,\theta)$, where $X$ and $Y$ are independent random variables, then $X+Y\sim ... 3answers 3k views ### Where is density estimation useful? After going through some slightly terse mathematics, I think I have a slight intuition of kernel density estimation. But I am also aware that estimating multivariate density for more than three ... 3answers 789 views ### Whence the beta distribution? As I'm sure everyone here knows already, the PDF of the Beta distribution$X \sim B(a,b)$is given by$f(x) = \frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1}$I've been hunting all over the place for an ... 1answer 756 views ### Why does MLE make sense, given the probability of an individual sample is 0? This is kind of an odd thought I had while reviewing some old statistics and for some reason I can't seem to think of the answer. A continuous PDF tells us the density of observing values in any ... 1answer 425 views ### Deriving Negentropy. Getting stuck So, this question is somewhat involved but I have painstakingly tried to make it as straight-forward as possible. Goal: Long story short, there is a derivation of negentropy that does not involve ... 1answer 17k views ### Pdf of the square of a standard normal random variable [closed] I have this problem where I must find the pdf of$Y = X^2$. All I know is that$X$has the distribution$N(0,1)$. What kind of distribution is$Y = X^2$? Same as$X$? How do I find the pdf? 3answers 5k 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 ... 1answer 31k 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? ... 1answer 194 views ### What is the name of the density estimation method where all possible pairs are used to create a Normal mixture distribution? I just thought of a neat (not necessarily good) way of creating one dimensional density estimates and my question is: Does this density estimation method have a name? If not, is it a special case of ... 5answers 22k 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 ... 2answers 68k views ### How to calculate the expected value of a standard normal distribution? I would like to learn how to calculate the expected value of a continuous random variable. It appears that the expected value is $$E[X] = \int_{-\infty}^{\infty} xf(x)\mathrm{d}x$$ where$f(x)$is the ... 5answers 2k views ### Probability that a continuous random variable assumes a fixed point I'm in an introductory statistics class in which the probability density function for continuous random variables has been defined as$P\left\{X\in B\right\}=\int_B f\left(x\right)dx$. I understand ... 1answer 9k views ### Intuitive understanding covariance, cross-covariance, auto-/cross-correliation and power spectrum density I'm currently studying for my finals in basic statistics for my ECE bachelor. While I think I have the math mostly down, I lack intuitive understanding what the numbers actually mean.(Preamble: I'll ... 1answer 7k views ### Density of Y = log(X) for Gamma-distributed X This question is closely related to this post Suppose I have a random variable$X \sim \text{Gamma}(k, \theta)$, and I define$Y = \log(X)$. I would like to find the probability density function of$...
I have a situation where I am able to estimate (the first) $k$ moments of a data-set, and would like to use it to produce an estimation of the density function. I already came across the Pearson ...
I have been given this task and was stumped. A colleague asked me to estimate the $x_{upper}$ and $x_{lower}$ of the following chart: The curve is actually a cumulative distribution, and x is some ...