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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24 views

Question about PDF's for Random Variables

"Suppose we have the random sample of n independent and identically distributed random variables with a normal distribution. The probability density function of each Xi is..." If our variable was e....
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19 views

Is $p(y|x) = p(y) = p(x)*|dx / dy|$; if $y = f(x)$? [on hold]

This is actually a two part question. for the first part, I do not understand why $p(y|x) = p(y)$ if $y = f(x)$ where $f$ is a deterministic known function of x. The source of confusion is the fact ...
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1answer
50 views

Calculating the sum of dependent uniform random variables

My question derives from Problem calculating joint and marginal distribution of two uniform distributions. So, suppose we have random variables $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as ...
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0answers
11 views

Mix pdf and cdf in binary response model [duplicate]

Let's suppose that I have a model that tells me how likely is for an event to have come after a certain time lapsed, given by some kind an exponential distribution, i.e. $$ \mathbb{P}(T_E < t) = \...
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28 views

Kernel Density Estimation - Comparison Between different sets of samples

Is there a way for compare the distribution of different set of samples? For example, I have three sets, for example: X1 = N(0, 1); X2 = N(0.5, 1); X3 = N(1, 1). Each set is drown with a specific (...
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1answer
27 views

Distribution of inbag matrix when sampling with replacement

Say I take a random sample of size $M$ from a sample of size $N$, like, for example you'd do when bootstrapping in random forest. As you increase $M$, you're more likely to sample any particular ...
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1answer
87 views

Find probability between $-\infty$ and $0$

The graph shown below is the numerically result of differences of Normal Distribution ($N(15.5 , 0.60^2))$ and Exponential Distribution $(\exp(0.5))$ (Both are independent). I am trying to find the ...
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1answer
112 views

Correct or Not? Probability and Geometry [closed]

A stick of length $1$ is broken into two pieces of length $Y$ and $1−Y$ respectively, where $Y$ is uniformly distributed on $[0,1]$. Let $R$ be the ratio of the length of the shorter to the ...
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1answer
36 views

CDF for f(x) = 0.5e^-|x|

This is the full question: "If a random variable has density f(x)= 0.5e^-|x|, for x∈R, find the cumulative distribution function". I know that to find cdf from the pdf you would have to integrate ...
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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 ...
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1answer
53 views

What information does a probability density function (PDF) graph provide?

This sounds like a simple question and I know PDF graphs are used a lot in presentations and financial publications. Yet, what information does it actually provide? The CDF actually gives you ...
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1answer
57 views

Probability density function for continuous random variable

The question might be very basic and stupid on certain levels, but please help me out here!! I recently picked up stats and went through discrete and continuous random variables. Discrete variables ...
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61 views

Convexity of conditional expectation

Define $g(k)\equiv\mathbb{E}(X|_{X>k})$ and assume that the probability density $f$ of $X$ is twice continuously differentiable. Is there a sufficient condition in terms of $f$ that imply that $g^{...
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22 views

The dynamics of a normal distribuition in stochastic processes (food court example)

Suppose I want model a huge food court. Let's assume that the number of people who start having a lunch is a function of time $f(t)$. Also, let's consider that the time people spend having a lunch ...
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16 views

Sampling from joint distribution by writing its density as a product of conditional densities

In Gelman et al. "Bayesian Data Analysis Ed3" the authors often do the following (e.g. on pg. 65): Given two parameters $\mu$ and $\sigma^2$ and data y joint posterior density $p(\mu,\sigma^2)$ is ...
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44 views

Quality of a quantile regression learner

Given a learning algorithm that selects and trains quantile models, how do we evaluate it? One idea is to - use the algorithm to train a model on a synthetic dataset with labels drawn from an ...
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14 views

Formalism to cope with probability density functions defined piecewise (in the second dimension)

I'm not sure how to pose this question, as I lack the correct terminology. Actually, my question tries to obtain insight on the terminology and notation to cope with the following problem: I have a ...
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1answer
39 views

Computing probability density function at a point, given the covariance matrix and mean

(Edited for clarity.) Say I have the variance-covariance matrix $\mathbf{V}$ and mean $\mathbf{\mu}$ of a multivariate normal distribution. Given a sample, $\mathbf{s}$, can I compute/estimate the ...
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2answers
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Intuitive explanation of “density generators”?

I was reading through Meucci's Risk and Asset Allocation (2005), when I happened upon the concept of a "density generator", which I have not been able to find good explanations for anywhere online, ...
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57 views

How to choose sample size from probability density for computing mutual information based on continuous variables

I need to compute mutual information gain based two continuous variables $X$ and $Y$ $I(X|Y) = \int_X\int_Y p_{x.y}(x,y) \log(\frac{p_{x.y}(x,y)}{p_{x}(x)p_{y}(y)})$. I have used Kernel Density ...
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Conditional density under conditional indepencence?

Let $X,Y,Z$ three random variables such that the joint density can be factorized as $$f(x,y,z) = f(x \mid z) f(y\mid z) f(z).$$ This is, I am assuming conditional independence of $X$ and $Y$ given $Z$....
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What does this assumption mean regarding equal marginal densities?

Suppose that we have a random variable $\epsilon$ with density $q(\epsilon)$ and $w = t(\theta, \epsilon)$, where $t$ is a deterministic function of a constant $\theta$ and random variable $\epsilon$. ...
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1answer
28 views

Estimating probability density function of big amount of data coming from MC simulations

I am trying to estimate Probability Density Function (PDF) of a big amount of data ($1e^6$ , $1e^7$, and higher) coming from Mote Carlo (MC) simulation. My objective is to estimate the PDF (e.g. with ...
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What does “density” really mean in probability density function in statistics? [duplicate]

I am familiar with the concept, but I simply can’t get my head over the intuition behind it. While being a derivative, it describes the rate of change for one unit. Simply put, we can say that it ...
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1answer
29 views

Computing a marginal distribution of a joint involving a delta function

Suppose that we have four continuous random variables $x,y,z,$ and $v$ and we want to compute the following integral: $$\int f(x\mid y)f(z\mid x,y)f(v\mid z,x,y)\,dx$$ There are a few conditions: $...
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33 views

Changing a conditional probability to a deterministic function

Suppose that we have a conditional density function $p(y|x;\theta^*)$, where $\theta^*$ represents distribution parameters and are assumed to be deterministic. Is it possible that we write this ...
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1answer
16 views

Two datasets with same length give different number of extremes

I have two datasets of a given variable x that have the same length, let's say 14600 values in total each one. I need to extract the extreme observations within ...
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30 views

Difference between characteristic function and F-transform

I'm struggling to understand the difference between this two functions. I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\...
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47 views

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 ...
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1answer
67 views

The pdf of the ratio of two lognormal distributions [closed]

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
23 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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42 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
42 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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25 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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39 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
108 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
72 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
17 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
32 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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14 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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1answer
71 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
192 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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19 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
44 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
53 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
61 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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63 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
33 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 ...