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Questions tagged [density-function]

For questions on using, finding, or otherwise relating to probability density functions (PDFs)

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1answer
18 views

Notation q(x) << p(x) in probability

I recently read an article on probability theory that use the notation: q(x) << p(x) where p(x) and q(x) are two density functions of two distributions. What does the operator << mean in ...
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10 views

Transition density of hidden Markov model

Background info: We have a target moving in $\mathcal R^2$ according to the model; $$X_{n+1}=\Phi X_n+\Psi_z Z_{n}+\Psi_w W_{n+1}, \quad n \in \mathcal{N}$$ The information contained in $X_n=\{X_n^1,...
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1answer
46 views

How to show the maximum likelihood of $\theta$?

Let $x$ have a uniform density $f_x(x\mid\theta) \sim U(0,\theta)=\left\{ \begin{array}{ll} \frac{1}{\theta} & 0 \leq x \leq \theta \\ 0 & \text{otherwise} \end{array} ...
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2answers
46 views

Finding $c$ when $X$ has pdf $f(x) = cx(1 - x)$ for $0<x<1$

Let $X$ be a random variable having density function as $f(x) = cx(1 - x)$, for $0 < x < 1$. Find the value of $c$. After solving this question I got the answer as $6$, whereas the answer ...
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1answer
35 views

Problem to calculate a marginal function in probability

I have a problem in probability. I have $f(x, y) = \frac 14 \cos(y) $, if $x$ is between $0$ and $\pi$, and if$ y$ is between $-\frac x2$ and $\frac x2$. I have to calculate $f(y)$. I calculated ...
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1answer
45 views

Finding the probability density function of Z=X+Y

I am stuck with finding the pdf for Z = X+Y, I have the pdf for X and Y. The problem: $f_Y(y) = 1/(b-a) ~~\big[25 \leqslant y \leqslant 35\big]$ $f_X(x) = 0.1 - 0.00667x+1.1\times 10^{-4} x^2$ I ...
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1answer
21 views

Probability Density Function without having the definite integral [closed]

So we're given a random variable $X$ and an $f(x) =\begin{cases} 0, & \text{if $x<0$} \\ cxe^{-x}, & \text{if $x≥0 $} \end{cases}$, where $c$ is a constant. How do I find the constant if ...
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42 views

How to prove that two random variables are independent

If $X$ and $Y$ are independent exponential random variables, find the joint density of the polar coordinates $R$ and $\Theta$ of the point $(X, Y )$ on 2-dim plane. Are $R$ and $\Theta$ independent? ...
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1answer
24 views

What is the distribution of a nested Laplace?

We have three random variables $X_1, X_2, X_3$ having the following conditional distributions: \begin{align} p(X_2 \mid X_1=x_1) &\sim \mathrm{Laplace}(X_2;x_1, b) = \frac{1}{2b} \exp(-\frac{|X_2 ...
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1answer
37 views

Determine independence of random variables by observing geometric shapes.

In the image provided there are 6 different shapes: If (X,Y) is uniformly distributed inside the area of each shape, are X and Y independent?. For how many of these shapes the answer is yes? For the ...
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1answer
31 views

Confusing Terminology: distribution or density?

This is a terminology question, which I am trying to understand. I have noticed when the term "distribution" is used, it refers to "cumulative" quantities (CDF) and when use "density" they refer to ...
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0answers
25 views

Joint density of a system of SDEs coupled through noise

I have a system of two (one-dimensional, Ito) stochastic differential equations, one describing the evolution of $X_t$ and the other the evolution of $Y_t$. The two SDEs are coupled through the noise,...
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0answers
33 views

How can I tell whether or not a function f can be a probability density function?

Consider the function: $ f(x)= \begin{cases} c(2x - x^3),& 0<x\lt \frac{5}{2}\\ \\ \;0 &\text{otherwise}\\ \end{cases} $ I need to determine whether or not this can be a probability ...
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2answers
27 views

probability with an absolute value given a probability density function

Having the p.d.f. $f(x)=\frac{1}{2}e^{-|x|}\mathbb{I}_{(-\infty,\infty)}(x)$ Determine the probability of $P(1\leq |x|\leq 2 )$ and what I did was \begin{equation} \begin{split} P(1\leq |x| \leq2)&...
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1answer
22 views

Density of a continuous distribution

For a discrete distribution, the density at a point would be the same as it’s probability, but for a continuous distribution (let’s say X exponentially distributed of parameter 1) we cannot do the ...
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Density function marginal distribution

I have to find marginal distribution for two-dimension variable. Density function for this variable is: 3/64( x1-x2)^2 to (X1,x2) from triangle (0,0), (-4,0) and (-4,4). I found the marginal ...
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1answer
36 views

Error in Proof for $E(XY|X) = XE(Y|X)$

Prove $E(XY|X) = XE(Y|X)$ $f_{X,Y}(x,y)$ be the Joint density of $X$ and $Y$ Then by Jacobian Transform, joint density of $XY,X$ is $f_{XY,X}(z,x)=f_{X,Y}(x,\frac zx)\frac1{|x|}$ $$E[XY|X]=\int_{\...
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18 views

a possible property of ordered pdfs

Suppose $f_1,f_2,\dots$ are pdfs of random variables with the same domain. Assume that $\frac{f_i(x)}{f_j(x)}$ is increasing in $x$ for any $i<j$. This is sometimes called the likelihood ratio ...
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1answer
18 views

Distribution of a probability density function

I have probability density function of the log normal distribution $f_X(x, \mu, \sigma^2)$, where $x$ is the random variable, $\mu$ is the location and $\sigma^2$ is the scale parameters of the ...
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1answer
13 views

Finding the value of constant from pdf

Consider a random sample $X$ with pdf $ f(x)=\frac{k(p)}{x^{p}} $ for $x>1$, $p>0$ and $k(p) $ is a positive constant. Find set of possible values of $p$ for which variance is finite and fourth ...
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1answer
14 views

$-\log(X)$ transformation of beta-distributed random variable $X$

Let $X \sim \text{Beta}_{(\theta, 1)} =: \mathbb{P}_\theta$ be a continuous random variable where $$\mathbb{f}_\theta(x) := \theta \cdot x^{\theta-1}\mathbb{1}_{[0,1]} = \cases{\theta \cdot x^{\theta-...
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1answer
81 views

If $f\in C^2(\mathbb R)$ is a probability density function, then $f'(x)\to0$ implies $f(x)\to0$ as $|x|\to\infty$

Let $f:\mathbb R\to[0,\infty)$ be twice continuously differentiable and assume $$\int f(x)\:{\rm d}x=1.$$ How can we conclude that as $|x|\to\infty$ either $f(x)\to0$ and $f'(x)\to0$, $f(x)\to0$ and ...
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1answer
53 views

Find conditional distribution of Y|T

Let $X_1 , X_2$ be iid random variables, following the pdf: $$ f_\theta (x) = \theta x^{\theta-1} $$ for $\theta >0$ and $0 <x<1$. Let $$T=X_1 X_2$$ $$ Y = \begin{cases} 1, &...
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11 views

Continous Observation with HMM & Gaussian

I've read that for HMMs with a discrete output we assign an emission probability $b_j(o_t)$ of emitting symbol $o_t$ when in state $S_j$. What I don't understand is why, in the case of continous ...
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1answer
19 views

Quotient Distribution of Positive Independent Random Variables

Suppose $X$ and $Y$ are independent positive random variables with probability density functions $f_X$ and $f_Y$ respectively. Show that $Z=X/Y$ is absolutely continuous and find its probability ...
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16 views

Joint density of two functions composed of independent draws

This is a more general way of asking my other more specific question. Suppose we have three i.i.d. draws, $(X_i)_{i=1,2,3}$ from some distribution with density $f$. Now, define two random variables $...
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23 views

How to calculate $\Pr(\gamma_2 < z | \gamma_1 \geq z)$, when the events are dependent?

How to calculate the following probability $$\Pr(\gamma_2 < z | \gamma_1 \geq z),$$ where $\gamma_1 \triangleq \frac{a_1g_1}{a_2 g_2 + 1}$, $\gamma_2 = a_2 g_2$, $a_1, a_2 > 0$ and $g_i, i \in \{...
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1answer
83 views

joint density of two sums of independent random var with common component

Suppose we have three iid draws from a uniform distribution on $[0,1]$. Call these random variables $A, B$ and $C$. Let $X=A+B$ and $Y=B+C$. I have figured out that the density of $X$ (or $Y$) is $$...
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28 views

2D disc-shaped region kernel density estimation boundary bias correction for Gaussian bandwidth $h$ and boundary radius $R$

I am trying to develop a closed form expression for the boundary bias correction factor for kernel density estimation in a circularly-bounded 2d region where Gaussian kernel diameter is $2h$ and ...
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14 views

Calculating density of individuals within an area (tree stand density)

Any help greatly appreciated! I want to find the density of trees surrounding a sample point (measured in trees/m$^2$). The distance from the sample point to the three nearest trees has been ...
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1answer
14 views

Density of “opposite” measure

Let $\mathbb{P}$ be a complete probability measure on the measurable space $(\Omega,\mathcal{F})$. Define the measure $\mathbb{Q}$ on measurable sets $A \in \mathcal{F}$ by $$ \mathbb{Q}(A)\triangleq ...
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22 views

Exponential Test

Given a collection of data points, without any other information I want to test for an exponential distribution. In a different case, I tested for normality using an Accord implementation of the ...
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Can a continuous time Markov processes have transition densities only at certain times?

Suppose that $X=(X_t)_{t\ge 0}$ is a Markov process on some state space $E\subset\Bbb R^n$ with transition semi-group $(P_t)_{t\ge 0}$. It is often nice to know, whether the transition measures have ...
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20 views

Notation of density functions

Assume two jointly normally distributed variables X,Y. Then what I often see is that densities are written as follows: $f_X(x)$ as the density of X, $f_Y(y)$ as the density of Y, $f_{X,Y}(x,y)$ as ...
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19 views

PDF of the time at which the second raindrops hits

Raindrops hit you at a rate or 1 raindrop per second, what is the PDF of the time at which the second raindrop hits you? Clearly, we have to use exponential random variable. Also we are asked to use ...
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2answers
51 views

Finding joint density function of two independent random variables

Let $X$ have density $f_X(x) = 2x$ for $x\in(0,1)$ and zero otherwise. Let $Y$ be uniform on the interval $[1,2].$ Assume that $X$ and $Y$ are independent. a) Find the joint PDF of $(X,Y)$. Use ...
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2answers
71 views

Sum of two continuous Uniform $(0$,$1)$ random variables without convolution

We can transform one continuous multivariate distribution to another based on two chosen transformation functions, their inverses and derivatives. These course notes go through the process in detail, ...
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0answers
50 views

Identifying a conditional distribution for Gibbs sampling

I have $N$ samples with $M$ features with class labels $T\in\{-1, 1\}$ which were generated by drawing each feature $m$ from a normal distribution $N\sim N(0, \sigma_m)$. Class labels were assigned ...
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1answer
43 views

Proof of $P(X<Y)$

Assume that $X$ is $Exp(\lambda)$ distributed and $Y$ is $Exp(\mu)$, and they are independent. I want to know how I can calculate $P(X<Y)$. I don't understand why $$ P(X < Y)=\int_{-\infty}^\...
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2answers
86 views

$P(Y \le X)=\int_0^{\infty} P(Y \le X | X=x)f_X(x)dx$

I was looking at a solution of a probability exercise and the author of the solution uses the formula $$P(Y \le X)=\int_0^{\infty} P(Y \le X | X=x)f_X(x)dx$$ where $X$, $Y$ are the random variables $...
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1answer
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Finding Cumulative distribution function of $f_x(X) = \frac{1}{4} \cdot |X|$

Let $f_X(x) = \frac{1}{4}\cdot |X|$ a probability density function. $-2<X<2$; Find the Cumulative distribution function of $f_X(x)$. What I thought was the answer is $F_X = \frac{1}2 +...
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66 views

Product of Normal and independent log-Normal. What is the density?

Let $X$ and $Y$ be independent standard Gaussian random variabless -- i.e., $N(0,1)$. Let $\sigma \in \mathbb{R}$. Define $$Z=X \cdot e^{\sigma \cdot Y}.$$ Is there a closed form expression for the ...
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1answer
36 views

Prove that $X$ and $Y$ are independent random variables.

Let $X$, $Y$ and $Z$ are random variables with density $$ f_{X,Y,Z}(x,y,z) = \frac{1}{\sqrt{\pi} \Gamma(\frac{n-1}{2}) \Gamma(\frac{n}{2})2^n}(xy-z^2)^{\frac{n-1}{2} - 1} e^{-\frac{x}{2}} e^{-\frac{y}...
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1answer
41 views

How to Find the CDF and PDF of Uniform Distribution from Random Variable

If a random variable is defined as $Y = 3 - 2X$, how do I find the CDF and PDF of $Y$ if $X$ follows a Uniform distribution of $X \sim (-1,1)$? For CDF, am I trying to solve for $F(b) = F(-2)$ or $F(...
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5 views

Bivariate density with uniform marginals

I kindly ask for your help to solve this problem. Consider two standard uniform random variables $X_1,X_2\sim U[0,1]^2$. Then, the questions are: 1) is it possible to find the explicit form of joint ...
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1answer
34 views

What makes a random variable absolutely continuous?

Consider this question: If $X$ is a random variable with $F_X$ (cumulative distribution function), then $X$ is absolutely continuous if: a) $F_X(x)$ is differentiable for all $x$ in the real ...
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1answer
45 views

Why is this function not a valid PDF?

My text book says that the following function is not a valid PDF: According to my calculation this could be a PDF if $k=3/2$ I applied integration to $f(x)$ between $1$ and $3$, and found $3/2$. ...
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0answers
10 views

Objective function to determine posterior distribution.

I am trying to understand the effect of the density of a random variable $a$ which itself is a function of several other random variables $a = f(x,y,z)$, the following question is in accordance with ...
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1answer
64 views

Gaussian Mixture Model: What is a “universal approximator of densities”?

When looking into Gaussian Mixture Models (GMMs), I encountered multiple times the statement that "GMMs are a universal approximator of densities" (e.g., [0]). I'm not sure whether I understand this ...
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2answers
29 views

Log of products and densities

By taking the log of the function: $$\prod_{t=1}^T n_t! \prod_{i=1}^{n_t} \frac{f_v[b_{i:n_t}]}{[1-Fv(r)]},$$ it is possible to end up with the following expression: $$ \log(n_t!)+\sum_{i=1}^{n_t} \...