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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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How to make sense out of integration over discrete data points?

Looking for a proof of the expected value of the score function equating zero, I came to this document that was recommended in another answer. Considering that we have a sample of n x_i values, I ...
0
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1answer
30 views

The pdf of the ratio of two lognormal distributions [on hold]

What is the pdf of the ratio of two independent lognormal distributions? Why is $log(X)$ normal when $X$ is lognormal?
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1answer
21 views

Roulette Wheel for sampling user defined pdf

Following is the pdf from which I want to sample so, I used roulette wheel sampling Code to generate pdf ...
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0answers
39 views

Finding the distribution of a piecewise function of a Gamma random variable

Let random variable $X \sim \text{Gamma}(\alpha,\beta)$. I want to derive the distribution of $Y$, where: $$ Y = \left\{ \begin{array}{ll} a X - k & \quad X \geq \frac{k}{a} \...
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1answer
39 views

Find PDF(X,Y) from PDF(X) and PDF(Y)

given that X and Y are not mutually exclusive, is there anyway to calculate PDF(X,Y) from PDF(X) and PDF(Y)? Following are a few plots made from the dataset. In above image i have to find how PDF(11,...
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0answers
20 views

Can I increase the sample size by generating random numbers to apply the Chi-Square Goodness of Fit Test?

Does increasing the sample size by random number generation change the distribution? I have a sample of size 8. Each sample value represents the number of bus arrivals at a bus stop every 15 minutes. ...
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0answers
36 views

Characterizing a distribution

I have a set of words which in a given year has a frequency of occurrence k. I can observe that these words follow frequencies k1, k2, k3,....kn in the following year. If I have some data in the form ...
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1answer
86 views

PDF of log transformed variable

I want to know if I've understood log transformation correctly in terms of functions of the distributions. If $\log(X)$ is normally distributed, then $X$ is lognormally distributed. Let's say I have ...
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0answers
8 views

Double selection with varying size selection set (beginner)

I'm self-taught in statistics, so I have some holes in my knowledge for sure. please bear with me. I have a hard time defining my problem. I want to figure out if my selection procedure (governed by ...
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1answer
67 views

what is difference between $95 \%$ CI of mean and 95% pdf of normal distribution?

We took sample mean $\mu = 14$, and $\sigma = 0.45$. Calculate required area of normal distribution We will apply $\int_{13.19}^{15} f(x) \ dx = 95\%$. 95 % CI of mean But if we took sample size ...
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1answer
14 views

Bound for density of random variable with finite second moment

Let $\mathbf{X}$ be a vector-valued random variable with finite second moment and density $\rho$. Assume that $\rho$ is bounded and continuous. As $\mathbf{X}$ has finite second moment, I hope to find ...
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0answers
29 views

Determining a probability distribution from constraints on where its mass is

Let $X$ be a random variable over the real line. Suppose that we know that $X$ is a Pearson distribution. Furthermore, suppose we know how the mass of $X$ is distributed into 6 intervals, so that if $...
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0answers
12 views

Balancing continuous covariates for oversampling

I'm currently looking into methods for restoring balance of a biased dataset with respect to a continuous variable. My problem is similar to this question, with the slight difference that I'm dealing ...
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0answers
24 views

Variance of Poisson distribution larger than mu?

So I made a program to calculate variance of Poisson distributions for different $\mu$ and wanted to assert than variance <= $\mu$, but noticed that for larger numbers the variance exceeded the ...
2
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1answer
69 views

pdf from a set of conditional pdfs

I have an interesting problem, i have seen in many text books ways of calculating conditional pdfs but not many where given a set of conditional pdfs for a variable we wish to calculate it's pdf. In ...
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3answers
180 views

Proving transformations of two independent chi-squared random variables is equivalent to a Beta distribution

I came across the following in some old class notes of mine: if $\chi_{v_{1}}^{2}$ is independent of $\chi_{v_{2}}^{2}$ then $\frac{\chi_{v_{1}}^{2}}{\chi_{v_{1}}^{2}+\chi_{v_{2}}^{2}}\backsim ...
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1answer
22 views

Integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$

I'm trying to integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$ using the fact that the integral of any normal PDF is 1. But I'm having trouble completing the ...
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0answers
18 views

Conditional transformation of variables

I've seen a trick for finding the p.d.f of $r(X,Y)$ where $X$ and $Y$ are r.v's by first calculating the cdf i.e $P(r(X,Y) \leq l)$ and then differentiating to find the pdf. So if $\Omega = \{(x,y) | ...
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1answer
42 views

Compute $E(X_1|X_1+X_2)$ $X_1, X_2$ both iid $Exponential(1)$

I recently stumbled across this question on CV: Conditional expectation conditional on exponential random variable And really liked the answer provided by @Rush, but I wanted to try to compute this ...
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1answer
50 views

If $X\sim\mathcal{N}(\mu = 1,\sigma = 4)$ find $\textbf{P}(X^2 - 2X \leq 9)$

If $X\sim\mathcal{N}(\mu = 1,\sigma = 4)$ find $\textbf{P}(X^2 - 2X \leq 9)$. I understand how to find the pdf of $X$, but I'm not sure how that would work for a function of $X$ like $X^2 - 2X \leq 9$...
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1answer
51 views

When a probability density function is defined to be finite?

In "Pattern recognition and machine learning" by Cristopher Bishop, Chapter 2.3.6 (pag. 100) says that The gamma distribution has a finite integral if $a>0$, and the distribution itself is ...
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0answers
52 views

How to pass from {Probability density function, convolution} to {Probability density function, characteristic function}?

In Forsman, W.C. (1986) "Polymers in solution: theoretical considerations and newer methods of characterization", Springer, New York. https://www.springer.com/la/book/9780306421464 page 24, it states:...
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1answer
28 views

What is the expected value of half a standard normal distribution?

You have a normal distribution with mean of 0 and variance of 1. Keeping the same probabilities and focusing only on half of the distribution (other half has it's original probabilities but x values ...
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1answer
85 views

What is a good name for a density function that does not relate to probability?

There is confusion between normalized functions whose area under the curve is one, i.e., density functions, and probability density functions that are not only density functions but that are measures ...
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0answers
15 views

Group comparison for bivariate distributions

For two groups A and B that consist of n and m individual samples. Each individual sample has a unique 2-dimensional joint probability density functions (PDFs)of two variables. These PDFs are ...
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0answers
21 views

How to approach the calculations of probabilities in high dimensions?

I don't really have too much trouble finding probabilities using joint probability density functions (PDFs) (of two variables) by drawing the area of support in the $xy$-plane, and then integrating ...
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1answer
108 views

Approximate density from moments and quantiles, then sample from it

Situation I need to send R code to a third party to run estimations for me (I will not be able to work with the data directly). I want to simulate data to test some of the estimators before sending ...
2
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1answer
36 views

wigner semi-circle distribution random numbers generation

I am trying to generate random numbers in Wigner semi-circle distribution. Since this one does not have the analytical solution for the inverse function of the pdf. I wonder if anyone familiar with a ...
0
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1answer
31 views

Expressing as a probability density function [closed]

The measuring error x is a normal random variable. Variance of the error = 4. If distribution of x can be shown by a probability density function f(x), how would you find the analytical expression of ...
0
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1answer
20 views

How to find emprical PDF by using the normalized histogram?

first of all, thank you for your time, here is my question; Is it possible to find emprical PDF by using normalized histogram? I am trying to learn discrete event simulation and what I see is there ...
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0answers
14 views

Mapping a range of values such that the resulting distribution is uniform [duplicate]

I have a set of values. Let's call the set X with values ... . Those values in [0, 1] have a non uniform distribution (empirically measured). I would like to re-map those values on [0, 1] such that ...
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2answers
75 views

Find mgf from joint pmf

The joint pmf of random variables $ X$ and $ Y$ is given by $$p_{XY}(x,y)= \begin{align} & \frac{e^{-2}}{x! (y-x)!}\quad\text{if}\,\,\,x= 0,1,...y,\ y=0,1,... \\ \end{align} $$ Find its mgf. \...
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1answer
31 views

An approximation to the cdf of the normal from a pdf?

In this paper (p. 36), authors wrote $$p(n,T) = \Phi \Big(\frac{n}{T},\mu,\sigma \Big) - \Phi \Big (\frac{n-1}{T},\mu,\sigma \Big)\; (3) $$ Bellow we will use the approximation $$p(n,T) =...
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1answer
23 views

Why is sample mean minus location parameter of shifted exponential gamma distributed?

My book says the following Suppose $X_{i} \sim$ iid $Exp(1,\eta)$ Where $Exp(\theta,\eta)$ is the shifted exponential ie has density $$\frac{1}{\theta}e^\frac{-(x-\eta)}{\theta}$$ for $x \ge \eta$ ...
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0answers
18 views

Variance Estimator for Density

Is there an estimator to predict the variance matrix of a 2D distribution given the value of its density sampled on a regular finite grid? I'm not even sure that estimator is the right word to use ...
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2answers
125 views

What is this “phenomenon” called?

Below is a histogram of some data, the bins are integers the other parameters are irrelevant. As you can see there seems to be two separate but overlapping normal distributions for odd and even ...
0
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1answer
40 views

Simulation: Generate random numbers that cluster around an average? [closed]

I want to simulate a simple event that has variable empirical result/outcome. Generate random numbers that cluster around an average For example, let's say we collect the data for how far people can ...
1
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1answer
53 views

Find $P(X<2Y)$ of a bivariate distribution

$f(x,y)=\frac{1}{2 \pi }\exp(-\sqrt{x^{2}+y^{2}})$ where $x,y$ in $\Bbb R$ My attempt: $$\Bbb P(X<2Y) = \int_{-\infty}^\infty \int_{-\infty}^{2y}\frac{1}{2 \pi }\exp(-\sqrt{x^{2}+y^{2}})\text{d}x\...
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0answers
101 views

Reparametrization trick for VAE, prooving that the resulting vector follows a normal distribution

So I've been reading about Variational AutoEncoders and I'm stuck on a little exercise meant to help understand the reparameterization trick. Z is a random vector of K elements with a distribution $q(...
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1answer
22 views

Probability question in Mat

My teacher give me this question: Using MATLAB, generate 10000 Random Vectors of size 500 with the PDF of Gamma distribution. Find the PDF of maximum and minimum of the generated Random vectors. (Use ...
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0answers
29 views

Substitution for unknown true density in 'Density Estimation Trees'

I'm having a hard time understanding parts of the derivation of the objective function for Density Estimation Trees (reference below) regarding the loss function. Taken from the article (Sec. 3.1): ...
0
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1answer
38 views

Little problem calculating hazard function

I'm starting to study maths again after a long time without having touch them and I'm currently with survival analysis. I want to get a hazard function h(x), and I know that it can be calculated as ...
0
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1answer
33 views

Problems calculating and plotting distribution function

After a long time without having touch anything related to maths or statistics, I decided to give myself another chance. I am currently refreshing some concepts of density and distribution functions, ...
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0answers
113 views

fitting non-normal multivariate distributions in R

I have many (n=317,823) observations on two variables. I want to fit a bivariate distribution to my observations, in order to identify descriptive features of the distribution (quantiles). However, my ...
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2answers
250 views

Fitting pmf of a scaled Poisson distribution and Python histogram plotting

I have a nuclei meanlife of $550\mu s$, for which I've taken the frequency(rate) to be $1/meanlife = 1818$. I then sampled randomly from a poisson distribution with that frequency, taking the ...
0
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1answer
14 views

Setting boundaries for calculating $P(Y/X>2)$ choosing $dx/dy$ order [duplicate]

Given two independent variables $X$ and $Y$, with marginal pdfs $f_X(x)=2x, 0 \le x \le 1$ and $f_Y(y)=1, 0 \le y \le 1$, calculate $P(\frac{Y}{X} > 2)$. So this can be written as $P(Y>2X)$, ...
2
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1answer
69 views

Finding the joint CDF using the joint PDF; why can't I do this?

Find the joint CDF of the independent random variables $X$ and $Y$, where $f_x(x)=x/2, 0\le x \le 2, $ and $f_Y(y)=2y, 0 \le y \le 1$. To do this, we can find the CDF separately for each of the ...
0
votes
3answers
78 views

Compute $P(Y<3X)$ using joint PDF

I'm given a joint pdf $f_{X,Y}(x,y)=2e^{-x-y}, 0<x<y, 0<y $ and asked to compute $P(Y<3X)$. To do this, I let $Y=3X$ (the boundary) and found that the region of integration is under this ...
0
votes
1answer
36 views

Order Statistics; Finding the probability that the first sample is < 0.6, and the last sample is > 0.6

Here is the problem statement below: A random sample of size 5 is drawn from the pdf $f_Y(y)=2y, 0\le y \le1$. Calculate $P(Y_1^{'} < 0.6 < Y_5^{'})$. Here, using formulas for order ...
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0answers
23 views

bandwidth setting for density comparison

I intend to compare a list of density distributions. Following one published paper (with similar type of data and same objective), I learned that I have to use Gaussian kernel density estimator, and a ...