Questions tagged [density-function]
For questions on using, finding, or otherwise relating to probability density functions (PDFs)
2
votes
1answer
31 views
Integration of probabilities - decision theory - minimizing misclassification rate
I have trouble understanding an equation in a book I'm reading. Basically,
Consider a decision rule that divides the input space into regions $R_k$ called decision regions, one for each class, such ...
1
vote
1answer
25 views
How to find CDF of $Y=|X|\wedge 2$ with $X\sim Laplace(\lambda)$
Given $X$ a bilateral exponential with density $f_X(x)=\frac{1}{2}e^{-|x|}, \forall x\in \mathbb{R}$ and $\lambda=1$, i have to find CDF of $Y=|X|\wedge 2$.
I know that $Y$ is not a monotonic ...
0
votes
1answer
31 views
Proof or relation between a Uniform and Exponential
Given $X\sim U(0,1)$, i have to determine the density of $Y=-\frac{1}{\lambda}lnx$.
I can't apply the law of transformation of random variables because $g(X)$ is not a monotonic function. So, i ...
0
votes
0answers
27 views
Probability Density Function Notation
So Im new to pdf's and am doing some preliminary research on the topic. I came across notation that I didn't understand. The problem I found said this:
A random variable I has the following PDF:
...
0
votes
0answers
11 views
How to show this distribution is proper?
This question is based on problem 14 from chapter 3 of Gelman et al.'s Bayesian Data Analysis.
We have four data points $y_i$ with covariates $x_i$, $i=1,\cdots,4$. We use the model: $$y_i\sim \...
1
vote
1answer
21 views
Proof of relation between Normal and Chi-square
Let $X\sim N(0,1)$, i want to determine the distribution of $Y=X^2$.
By definition, the density of $X$ is $f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2},\forall x\in \mathbb{R}$...
1
vote
2answers
24 views
Discretization of an exponential variable
Given $X=Exp(\lambda)$, i have to define $Y=ceil(X)$ in order to prove the link between exponential and geometric variables.
By definition of ceiling $\forall x\in \mathbb{R},\exists n\in \mathbb{N}:...
0
votes
1answer
20 views
Calculating the density function of a variable based on a relation [on hold]
i was wondering how to get density function from a mass function for another density function
-1
votes
0answers
12 views
questions on conditional probability [closed]
I don't know which statement below in the link of picture is right. Though I learn the conditional probability, I'm quite confused and have know idea how to tackle those question. which one is right?
1
vote
1answer
25 views
Cumulative function of exponential bilateral function
I'm trying to calculate the cumulative function of a Laplace distribution whose PDF is
$f_X(x)=\frac{\lambda}{2}e^{-\lambda|x|},\forall x \in \mathbb{R}$
I proved that, if $x<0$:
$\int_{-\infty}...
0
votes
1answer
27 views
Casorati-Weierstraß theorem - dense
I try to understand the Casorati-Weierstraß theorem. But I don't understand when a picture is dense in C. $e^{1/z}$ is dense, $1/z$ isn't. But why?
Thanks.
0
votes
0answers
27 views
What is the conditional probability density $f_{X|Y_1}(x|y_1)$ if $y=G(x)$?
Let's say we have random variables $X$ and $Y$, $x,y \in \mathbb{R}^n$ with densities $f_X(x)$, $f_Y(y)$, and we also have $y = G(x)$ (where $G$ is a bijective, differentiable function). I know that $...
0
votes
1answer
61 views
Marginal Density Function, Gamma and Beta distributions
If $Y\sim\operatorname{Gamma}(\gamma,\delta)$ and $Z\sim\operatorname{Beta}(\alpha,\beta)$ then their density functions are, respectively,
$$
f_Y(y)=\frac{\delta^\gamma}{\Gamma(\gamma)}y^{\gamma-1}e^...
4
votes
2answers
48 views
What will be the pdf of $X+Y$ if $X$ and $Y$ are iid from Cauchy? [duplicate]
Suppose $X$ and $Y$ follow Cauchy distribution independent of each other. What will be the pdf of $X+Y$?
What I got by using convolution theorem is that the density $g$ of $X+Y$ is $:$
$$g(x) = \...
0
votes
1answer
32 views
What will be the pdf of $X+Y$?
Suppose $X$ and $Y$ are independent random variables. Let $f$ and $g$ be the pdf of $X$ and $Y$ respectively. Let $h$ be the pdf of $X+Y$ then can we say that $h(x)=f(x)g(x),$ for all $x \in \Bbb R$? ...
-1
votes
2answers
18 views
joint density of gaussian distributed x,y. show that z = (X, Y) is not a gaussian vector [on hold]
here's a joint density of x,y. i'm asked to show that each of them has gaussian distribution, did it.. each of them distributed N(0,s^2) (s for sigma)
enter image description here
now i'm asked to ...
0
votes
1answer
17 views
Finding probability, when density function is given
I have density function $f(x,y)=4xy \mathbb{I}_{[0,1] \times [0,1]}(x,y)$.
I need to find:
1) $\mathbb{P}_F([0,1/3]\times[0,1/2])$
2) $\mathbb{P}_F([2/3,1]\times\mathbb{R})$
So in 1) I got $1/36$ ...
1
vote
1answer
40 views
PDF of combination of i.i.d. standard normal variables.
I assume that $a_i$,$b_i$, and $c_i$ are independent standard normal variables by each other ($a_i$, $b_i$, $c_i$ ~ $N(0,1)$).
and equation which I want to get PDF or mean and variance is $$\frac{\...
0
votes
1answer
25 views
How to get distribution function
I have a density function $g(x,y)=4xy \mathbb{I}_{[0,1]\times[0,1] }(x,y)$. How can I get a distribution function $G(x,y)$?
I found that $G(x,y)= \int _{-\infty}^x \int _{-\infty}^y f(u,v) du dv$ ...
3
votes
1answer
42 views
Let $X$ and $Y$ be jointly continuous random variables. Find an expression for the density of $Z=X-Y$
This question was from the problem section of my textbook but I'm not sure how to start it.
At first, I thought it would just be like finding the sum of two variables which can be found in the ...
0
votes
1answer
23 views
Multivariate random variable normalization PDF proof verification
It would help me if someone can verify the following two proofs I made for the statement below (it's regarding only the second proof).
Let us have a unit ball centered at $(0,1,0)$. And let $(X,Y,Z)$ ...
0
votes
0answers
12 views
Sequence of convolutions
I am trying to find $F_k(t)$ defined by
$F_k(t) = \int\limits_0^\infty F_{k-1}(t-y)dF(y)$ and $F_1(t) = F(t)$ for the probability density function $f(t) = \begin{cases} \rho e^{-\rho (t- \delta)} \...
0
votes
1answer
32 views
Joint density function of $X$ and $X-Y$, where $X, Y\sim U(-1,1)$
Let $X$ and $Y$ be independent random variables following $U(-1,1)$. Find the joint CDF of $U=X-Y$ and $V=X$.
I found the Jacobian of the transformation to be equal to $1$, and $f_Xf_Y=\frac{1}{4}$. ...
0
votes
0answers
8 views
Kernel density estimation with knn
I would like an example with real data on how to calculate kernel density estimation with k nearest neighbor and how to make a density plot.
0
votes
1answer
41 views
Normalization transformation of a probability density
Let $(X,Y,Z)$ be distributed according to the pdf $p(x,y,z)$. What would be the pdf $q(x,y,z)$ of the multivariate random variable $(X',Y',Z') = (X,Y,Z)/\sqrt{X^2+Y^2+Z^2}$? The method that I have ...
-2
votes
1answer
23 views
How to use probability density function on this given scenario? [closed]
An officer is always late to the office and arrives within the grace period of ten minutes after the start. Let X b the time that elapses between the start and the time the officer signs in with a ...
-1
votes
1answer
25 views
How to calculate expected value and variance of a random variable [closed]
Let the random variable $Y$ have the following density:
$$f(y) = \frac{1+\beta y}2, -1 \le y \le 1, -1 \le \beta \le 1$$
Find $E(Y)$ and $V(Y)$.
Can anyone help me with this problem?
0
votes
0answers
30 views
Compute the density, distribution function
Compute the density function of Z = X2/X1 given that X2 and X1 both have the normal distribution with mean 0, variance 1 as their density.
I got this far:
$$
F(z) = P( X_2 \leq X_1 z) = \int\int \...
1
vote
1answer
42 views
Proving Cauchy random variables
Trying to prove that if a random variable $T$ has Cauchy distribution
with probability density function: $$f(x)= \frac{1}{\pi(1+x^2)}$$
then $X = \frac{1}{T}$
and $Y = \frac{2T}{1-T^2}$ are also ...
0
votes
0answers
17 views
Limits of Integration When Finding Marginal PDF
My question is about the same exercise problem asked here. I had a different question though regarding it, and decided to ask my own question.
More specifically, I keep seeming to have trouble ...
0
votes
1answer
19 views
Double integrals - how are the boundaries chosen?
I was looking at the proof of the theorem stating that for two independent rv's $X,Y$ with density functions $f,g$ the density function of the rv $X+Y$ is the convolution of $f$ and $g$.
After ...
1
vote
1answer
37 views
Finding density function for a given distribution
Let $X$ be a random variable with distribution function
$$
\\ F_X(t)=\begin{cases} 0, & t<0 \\ 2/11, & 0\leqslant t<1 \\7/11, & 1 \leqslant t<2 \\1, & 2 \leqslant t \...
0
votes
1answer
22 views
Integration of PDF's to find valid constant
I'm studying probability theory and came across and exercise problem that I would like to ask help with. Here's the specific problem that is in question:
Let $X$ and $Y$ have joint PDF
$$f_{X, Y}(x, ...
0
votes
1answer
11 views
Generalised (general) Uniform Distribution (continuous)
I have seen the Uniform Distribution/a uniform random variable for some interval in $\mathbb{R}$. For example $U(a,b)$ has probability density function $\frac{1}{b-a}$ (noting this is the 'volume' of ...
0
votes
0answers
10 views
K Nearest Neighbor Density Estimation Example on a sample
I would like to have an example on how to apply K Nearest Neighbor Density Estimator.
Although I understand the math I can't implement it in a sample to understand it visually.
1
vote
1answer
67 views
Integral of pdf and cdf normal standard distribution
$$ \int_{-\infty}^{\infty}\Phi(a+bx)\phi(x)dx=\Phi\left(\frac{a}{\sqrt{1+b^2}}\right) $$
I have a problem with showing the above result, where $\phi(x)$ and $\Phi(x)$ respectively are the pdf and cdf ...
0
votes
0answers
21 views
Uniformly Distributed Marginal Density [duplicate]
A point $(X,Y)$ in the Cartesian plane is uniformly distributed within the unit circle
if $X$ and $Y$ have joint density:
$$f(x,y) =
\begin{cases}
\frac{1}{π} & \mathrm{x^2+y^2} \le 1 \\
0 &...
0
votes
1answer
14 views
Finding the Joint CDF of a Continuous Density
I am working on a problem and am a bit confused:
The problem:
Find the joint CDF of the function:
$$
\begin{split}
p(x,y) &= {1\over8} (x+y)\\
0 \le X &\le 2\\
0 \le Y &\le 2\\
\end{...
0
votes
0answers
16 views
Evaluating probability density functions for a random variable x
How do I go about finding (0.2,0.3] because the interval is not inclusive. Also is my solution for ii) correct?
Thank you again! New at these types of problems and trying my very best
0
votes
2answers
23 views
What are the properties of this cumulative density function
I'm given a probability density function $\lambda e^{-\lambda x}$, I therefore deduce that the cumulative density function is its ingegral: $\int -exp(-x\lambda)$
I'm trying to find out the ...
1
vote
1answer
20 views
Marginal PDF from joint PDF
$$f_{X,Y}(x,y)~=~\begin{cases}2 & if &0< x < 1 ~,~ 0< y< x \\[1ex] 0 & \text{otherwise} \end{cases}$$
Now I should calculate the marginal PDF's.
So I use integral and then I ...
0
votes
2answers
32 views
Difficult integral and joint probability (calculating something like $Pr(X \geq b-a, X \geq Y-a)$) where $X$ and $Y$ are random variables
I'm trying to have some practice with joint probabilities, and I came up with a question that I'm struggling to answer.
Suppose that we have two random variables $X$ and $Y$, where $X$ is ...
0
votes
1answer
32 views
Sketching a region of a joint density function
This problem is from the textbook by John Rice, "Mathematical Statistics and Data Analysis":
Let $X$ and $Y$ have the joint density function
$f(x,y)=k(x-y),$ for $0\le y\le x\le 1$ and $0$ elsewhere
...
0
votes
0answers
16 views
Parametric Estimation with Different Distribution Populations
I am looking for some reference for a parametric estimation problem with different populations. Suppose the parameter we are interested in is $\theta$, and we have data, say, from two probability ...
1
vote
1answer
17 views
Easy integral,density,definite integral
I cannot compute the following integral. I'm confused with upper and lower bounds which are somehow switched from my point of view:
$$\int_0^\infty\lambda e^{-\lambda x}dx=1.$$ This is supposed to be
...
0
votes
0answers
31 views
Generating vector inside a $n$-sphere
I want to generate k n-dimensional vectors which are all inside a r-radius n-sphere and the most important : I want something uniformly distributed inside the n-sphere.
My initial idea is to generate ...
0
votes
2answers
30 views
The set {${m^m}/{n^n}: m,n\in \mathbb{N}$} density in ${\mathbb{Q}_+}$
I missed this topic (density) , so need help.
Is the set {${m^m}/{n^n}: m,n\in \mathbb{N}$} everywhere dense in positive rational numbers set?
I think maybe first of all here we must try with, for ...
1
vote
0answers
18 views
Integration of the product of PDF and scaled CDF of the Normal distribution
I am trying to calculate the integration
$$\int_C^1 \phi(x) \Phi \left( \frac{C}{x} \right)dx$$
where $\phi(x)$ and $\Phi(x)$ are the PDF and CDF of the standard Normal distribution respectively; $0 &...
0
votes
1answer
32 views
How to determine marginal density function
I am a beginner in statistics, and am self-studying. I have trouble understanding the marginal density function. I was wondering if any of you could please kindly help me with this problem.
How can ...
2
votes
2answers
77 views
Let $X,Y $ be two independent random variables with exponential distribution and parameter $\lambda > 0$.
Let $X,Y$ be two independent random variables with exponential distribution with parameter $\lambda > 0$. Let $S = X + Y$ and $T = \frac{X}{S}$. I want to find the joint density function of $(S,T)$...
