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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Coin flipping example (Bernoulli distribution) [closed]

I have a problem with two coins. One of them is an ordinary coin, while the other coin favors the heads side by a factor of 2/3. "U" is the total of observed outcomes and "V" is a ...
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Probability of a vector. Is my notation correct?

Question 1: Let us first consider the univariate case: Suppose we have $Y\in\{0,1\}$, suggesting that $Y$ is Bernoulli variable (and hence discrete) and $X\in \mathbb{R}$, then we know that \begin{...
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3 answers
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Alternative formula for the Bernoulli pmf?

If I understand correctly, a Bernoulli pmf just needs to assign a probability $p$ if there is a success $(x=1)$, and $1-p$ otherwise $(x = 0)$. Rather than the usual formula, can't the following ...
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Deriving distribution under change of variables between spaces of unequal dimension

For a function of random variables $T:\mathbb{R}^n \mapsto \mathbb{R}^m$ Wikipedia outlines how to handle three cases: $m = n = 1$ $m=n > 1$ $n>1 \land m=1$ There seems to be two missing cases:...
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What is the probability density function of a parallelogram [closed]

A very short question: What would be the probability density function of a parallelogram? Could we consider it as a two triangular distribution (in pink) that behaves the same and a uniform ...
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1 vote
1 answer
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How to interpret height in probability density function? [duplicate]

Assume I have a continuous variable X with PDF f(x). I know that, for every value of x, P(X=x) = 0, so the value f(x) is not the probability. But what is it exactly? If we have f(a) = 2 * f(b), can we ...
3 votes
1 answer
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PDF of $Z=X^2 + Y^2$ where $X,Y\sim N(0,\sigma)$

Using normal distribution probablilty density function(pdf), \begin{align} f_Y(x) = f_X(X) &= \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{x^2}{2\sigma^2}} \\ \end{align} Taking $Z' = X^2 = Y^2$, the ...
2 votes
0 answers
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Derive the pdf/cdf of a variable given as a formula of two random variables [closed]

Let's assume I have two random variables X and Y with x>0 and y>0, respectively. Let's also assume that their marginals are known as well as the joint cdf is known as the product of the two ...
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Fast measure of "clusteredness" of points?

I have a cloud of points in a bounded volume in 2D (lets say 2d for now, though it'd be nice to generalize to any dimension): $<p_n \in \mathbb [0, 1]^2: n \in [1..N]>$ I'm looking for some ...
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Complete workflow for best distribution fits

I'm studying streamflow data and i want to: (1) find best distribution for Annual (and montlhy) Q7 (minimum mean of 7 days flow per year) series; (2) obtain Goodness-of-fit for the best distribution ...
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Evaluating/combining PDFs over time to predict future value

I am trying to predict a value over time. I have historical data that I have used to calculate PDFs for the change over various time intervals. If I'm trying to predict the value at time T0 and start ...
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I need help understanding why this integral is the probability for winning by switching in the Monty Hall problem

I need help understanding this probability from the Monty Hall problem. Why does this integral give the probability of winning by switching if the Car is behind 1, Monty shows goat behind 3 and Player ...
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How can I derive the distribution of the L2,1 norm if the ditribution of L1 norm is given?

I understand that the L1 norm promotes sparsity and is a Laplace prior in the LASSO regression framework. I am interested in how this prior changes when we apply L2,1 regularisation instead? Is it ...
1 vote
1 answer
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Can CDF of a real random variable be a complex function? What does it mean physically?

I have a random variable $X$ which follows the following probability density function, $$ p(x) = \frac{1}{4\pi} \Big[ \operatorname{erf}\Big(\frac{k\mu-x+2\pi}{\sqrt{2}k\sigma}\Big) - \operatorname{...
7 votes
2 answers
326 views

How can I fit distribution for data which "almost fits"?

I have a sample for events occurring at certain continuous distances (kilometers), let's suppose emergency calls to hospitals. I have 200k observations, coming from 500 hospitals for an entire month. ...
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Differences between Bhattacharyya distance and Jensen–Shannon divergence

Based on Wikipedia, both the Bhattacharyya distance and Jensen–Shannon divergence are a method of measuring the similarity between two probability distributions. But now curious about - how are they ...
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Quick question about density with respect to product measure

Suppose we got 2 non indepenendent random variables $X_1:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_1,\mathcal{B}_1)$ and $X_2:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_2,\mathcal{B}_2)$ with ...
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1 answer
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Would this way of evaluating this probability be correct?

Suppose I have a discrete variable $S_t$ and a continuous variable $X_t$. Further, suppose I wish to evaluate $P(S_t=s_t)$. Would the below derivations be correct? \begin{align} P(S_t=s_t)&=\int P(...
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Probability density function of a zero-inflated (or zero-one-inflated) beta distribution?

I am interested in plotting the PDF of a zero-one-inflated beta distribution so that I can overlay an empirical density function of the observed data, with a PDF using the parameters I estimated from ...
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Estimating density function at specific percentile using empirical cdf

Summary I'm running an experiment where I'm using the empirical CDF of a known random variable to approximate the density of the random variable at a specific percentile. In general I'm interested in ...
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Mapping Parametric Curves with auxiliary variables

The image below displays an approach of using an auxiliary variable to map the parametric curves of a standard normal pdf and cdf. In Equation (1), z as r.v. is clearly one-dimensional. However, after ...
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Integral over subset of parameter space in which densities of parameters limit to 0

Let $\Theta \subseteq \mathbb{R}$ denote the space of a parameter $\theta$. Also define the subsets $\Theta^* \equiv \left\{\theta: c - \varepsilon < c < c + \varepsilon\right\}$ and $\Theta^+ \...
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How to interpret the values retuen by probabiltiy density function? [duplicate]

dnorm() and dgamma() are used to return the density of a given distribution function. How we interpret their value? For example, ...
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4 votes
2 answers
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How do you recalculate the probability density function if some time has passed and the event has yet to occur?

Suppose I have a probability density function for an event that is certain to occur between $0$ and time $T$, $p(t)$. However, some time $t_0$ has passed and the event has yet to occur. So I would ...
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Can the multivariate-t distribution have density greater than 1? [duplicate]

I'm working with a Python implementation of the Multivariate T distribution, and I've noticed when I evaluate the PDF at certain points, the likelihood returned is > 1. This is causing issues in ...
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Esitmate of minimal of a function changed after transforming the variable

I want to perform MCMC or HMC for solving minimization problem of a function $f(x)$, then define the corresponding density $$g(x) = \exp\left(-f(x)\right)$$ Because the function of the future apply is ...
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PDF of the linear combination of two order statistics

I have just started to learn order statistics so please feel free to correct my notation/terminology. In my field is common to provide data as the median of some sample (for example several ...
5 votes
2 answers
48 views

Normal density's rate of convergence to 0 as mean goes to infinity while x and standard deviation are fixed

Consider the density of the Normal distribution given by $$f(x; \mu, \sigma) = \dfrac{1}{\sigma\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{x - \mu}{\sigma}\right)^2\right)$$ It is obvious that, ...
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Area between two probability density functions as distance measure

I have two distinct probability density functions, and I would like to find a synthetic measure of how different the two distributions are. Intuitively, it would make sense to me to compute the area ...
2 votes
1 answer
37 views

What does evalue the density of a fitted model at specific points mean?

I have read a description of R package and find the following: "Evaluate the density of the fitted model at (2.747, 0.1467, 0.13, 0.05334)". I do not understand what the author mean by ...
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MLE is undefined for "densityless" distribution like Cantor distribution

I thought of a situation where we are given a random variable $X$ that has a Cantor "rescaled" distribution. That means that for a parameter $p>0$, $X$ has CDF $F_X(x)=C(\frac xp)$. This ...
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3 votes
1 answer
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Can a sequence of PMFs converge to a PDF?

Is there a meaningful sense in which a sequence of PMFs (of a corresponding sequence of real-valued random variables) can uniformly converge to a PDF? Intuitively, it seems like a strange question to ...
1 vote
1 answer
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Is sequence of probability mass functions always uniformly bounded

Say that we have a sequence of discrete random variables, $\left\{X_n\right\}_{n \in \mathbb{N}}$, which converges to a random variable, $X$, with a continuous distribution, e.g., the Normal (Gaussian)...
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1 answer
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Finding the conditional distribution from given normal distributions using Bayes' theorem

Background This question is related to my previous question: Describing the measurement of a random variable as another random variable, but I've narrowed and clarified my question. I think I've ...
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Mean squared error of 2 distributions when there are many zero values

I am comparing two (2-D) distributions using mean squared error. I take the difference between distributions, square them, average them and multiply them by the area of a pixel/bin. These ...
5 votes
0 answers
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What's the distribution of $|y-z|^2/|y-\bar{y}|^2$ for vectors with i.i.d. standard normal coordinates?

Let $y_1, y_2, \ldots, y_n$ and $z_1, z_2, \ldots, z_n$ be samples of size $n$ of a normal distribution $\mathcal{N}(0,1)$. My goal is to find the distribution of $$\frac{\sum_{i=1}^n (y_i - z_i)^2}{\...
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ML algorithm for distinguish between different activities

I have a simulated dataset which includes different activities like running, walking, eating etc now i want to apply ML algorithm to find how that algorithm can be used to distinguish between ...
3 votes
3 answers
215 views

Distribution change of variables

I'm working on distributions for a physics problem and I am quite stumped on how to proceed properly. The problem is as follows: I start at the point $(0,0)$ and go a distance $\eta$ in x direction $(\...
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1 answer
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Is there a meaning to the integral of $x \times f(x)$ over a range that is not infinite?

I know that the expected value can be computed as : $\mathbb{E}(X) = \int_{-\infty}^{\infty}xf(x)dx$ What if we do not do the integral over the whole range but only up to some value? Would there be ...
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How to prove symmetry of a Uniform kernel?

I am trying to prove this kernel is valid, $$ K(x) = \frac{1}{2}I(-1 < x < 1) $$ So far I can integrate to 1, but how do I prove $$k(x) = k(-x)$$ Also, how do we satisfy that k(x) is $\ge$ 0 for ...
0 votes
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Question Evaluating PDF of Transformed R.V

I'm curious if the following method is valid, or if I need to use the gradient of the derivative as is typical when computing the pdf of a transformed r.v: Let $g(\theta) \sim f$, where $g(\cdot)$ is ...
3 votes
2 answers
53 views

How to go from this formula to F distribution

I have the following formula. $$\int_{0}^{\infty} x^{\frac{n-1}{2}-1} (a+x)^{\frac{1}{2} \frac{-n}{2} -v} dx.$$ The quantities $d1 ,d2$ appearing in the pdf of the F distribution are the following. $$...
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1 vote
1 answer
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Estimate at which point a linear model hits a certain value (including probabilities)

I have a simple 1D set of datapoints with a trend, I want to estimate at which point $X_t$ (i.e., at which point in the future) the model will hit a certain threshold $Y_t$: I can fit a trendline to ...
0 votes
0 answers
76 views

Estimated lognormal PDF shifted compared to data

I have experimental data (~100k observations) that appear to be from a lognormal distribution. I am attempting to estimate the distribution parameters using ...
2 votes
1 answer
81 views

The probability/cumulative density function for inequality of two random variables

I have two random variables X and Y which came from different inverse gaussian (IG) ...
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1 answer
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Find the MLE density function of uniform [-\theta,\theta] [duplicate]

For $X_1,\dots,X_n$, i.i.d $X_n \sim \mathrm{unif}[-\theta,\theta]$, the ML: $\hat\theta_{MLE}=\mathrm{max}\{-X_{(1)},X_{(n)}\}$. Find the density function. Hint: For $x_1,\dots,x_n$ : $\textrm{max}\{-...
1 vote
0 answers
65 views

What is the probability mass function of Rock, Paper, Scissors?

I was curious about the statistics behind the game of rock, paper, scissors. Let's say n people are playing, where n is greater than or equal to 2. If when all n people reveal their play and only 1 ...
5 votes
2 answers
463 views

Is the probability of a continuous variable obtained via integrating over an interval of the probability density curve *cumulative* probability?

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The distribution function of appropriately normalised trimmed mean $T_n(\hat{F}_n)$

A definiton of functional of trimmed mean: $$ T^{\alpha}_n = (n-\lfloor \alpha n \rfloor)^{-1} \sum_{\lfloor\alpha n\rfloor + 1}^{n-\lfloor \alpha n \rfloor} X_{(i)}$$ A definition of the optimal ...
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2 answers
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Question on solution of Casella and Berger Exercise 9.10: Showing that $Q(t,\theta)$ is a pivot

My question concerns Exercise 9.10 of Statistical Inference by Casella and Berger: On page 428 the authors say In general, suppose the pdf of a statistic $T$, $f(t|\theta)$, can be expressed in the ...

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