All Questions
Tagged with density-function integral
47 questions
0
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54
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Deriving the pdf of noisy signal: sum of pdf f(s)=1/2+s/2 and a uniformly distributed noise [duplicate]
I am having a hard time to find the pdf of $\widetilde{s}=s+x$.
$s$ has a pdf $f(s)=1/2+s/2$ where $s\in(-1,1)$. $x$ is uniformly distributed over $[-\epsilon, \epsilon]$. I am trying to use the ...
0
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0
answers
27
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What is the pdf of the integral of a gaussian process and of the ratio of two gaussian variables?
I need to evaluate the moment functions of a zero mean gaussian process that constitutes the mathematical model of the seismic ground acceleration during an earthquake.
0
votes
0
answers
25
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Question on the proof step in the theorem 1 of the Gap statistic paper
From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422),
$\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
- 303
0
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0
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48
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Converting an integral into a probability of some event
Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$
$$$$Consider the ...
- 217
1
vote
1
answer
90
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Any closed-form solution to this integral (multivariate exponential)?
Here is the probability density function (unnormalized) of a covariance matrix: (from a Bayesian perspective):
$$
f(\boldsymbol{V})\propto \det(\boldsymbol{V})^{-\frac{N+J+1}{2}}\int_{\mathbb{R}^{K}}\...
- 71
1
vote
1
answer
63
views
How to combine two integrals containing the PDFs of a variable and its linear transform?
Original Post:
Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$.
I want to compute $Z(x) = F(x) - G(y)$.
What I have ...
- 13
3
votes
1
answer
177
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Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian
Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$.
I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
- 33
3
votes
1
answer
91
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From density function to cumulative distribution function?
Given $$f(y)=\theta/(\delta^{\theta}y^{\theta+1})\mathbb{1(y>1/\delta)}$$ where the last factor is the indicator function, and I am asked to compute the Cumulative Distribution Function of y:
$$\...
- 65
3
votes
1
answer
201
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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 ...
- 71
9
votes
2
answers
1k
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Probability of collision: mathematical vs probabilistic modeling
$\newcommand{\icol}[1]{% inline column vector
\left(\begin{smallmatrix}#1\end{smallmatrix}\right)%
}$
Scenario:
Let's consider a road segment on which there is continuous flow of cars circulating at ...
- 317
2
votes
1
answer
52
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About transformation of Variables
In a probability chapter of a Python Book, there is the following problem involving a transformation of variables:
I don't fully understand where the value 1/z+1 in Y > X(1/z+1) comes from, and ...
user309075
0
votes
1
answer
71
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Expectation of ratios of probability density functions
I'm trying to solve/simply the expression below:-
$\large \mathbb{E_{x \sim b(x)}} B\ [log\left(1 - \frac{A\ a(x)}{2\ c(x)}\right)]$,
or
$B \large \int_{x}b(x)log\left(1 - \frac{A\ a(x)}{2\ c(x)}\...
- 33
3
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0
answers
56
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Integral of difference of density functions of two Continuous Random Variables goes to 0
The problem says :
Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
- 619
1
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0
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47
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How do I evaluate this Integral?
I’m reading Glen Cowan’s book on statistical data analysis and was stuck on an integral that has to do with crv transformations. I’m not able to evaluate the integral
$$
g(a)da = \int_{x(a)}^{x(a)+|\...
- 11
2
votes
1
answer
202
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Help with an application of Leibnitz's Rule
I'm trying hard to understand and solve the following:
$$f_Y(y)=\frac{d}{dy}F_Y(y)=\frac{d}{dy}\int_{-\sqrt{y}}^{\sqrt{y}}{f_X(x)}dx=?$$
The background information is that $f_X(x)$ is the pdf of ...
- 697
7
votes
1
answer
330
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Expressing a marginal probability using copulas
Please correct me if I am wrong and kindly provide me with the correct notations. I have two questions:
We know that for the variables $(X,Y,Z)\in \mathbb{R}^3$, the marginal joint density $f(x,y)$ ...
- 1,226
3
votes
1
answer
649
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Finding C for which f(x) is a density function
One of the points of the exercise states:
Find the constant $C$ for which the following function is a density function
$$
f(x)=
\begin{cases}
C(x-x^2) & 0 \leq x \leq 2\\
0 ...
- 33
1
vote
0
answers
47
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Integrate out the binary indicator variable in a two-sample ANOVA
I have two sets of data, A and B, that have unequal sizes, and I want to compare their means. The standard approach would be to do a t-test. Getting a little more sophisticated, we can think of that t-...
- 67.2k
1
vote
1
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184
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What is the analog of the PDF and CDF for the likelihood function?
In probability, we can find the cdf using the pdf and vise-versa.
Integrating pdf yields the cdf.
Does integrating the likelihood function yield any important thing?
In statistics, $\mathcal{L} (M\...
user271077
2
votes
1
answer
224
views
Find conditional pdf given joint
Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$.
What is the marginal cdf of $X$? of $Y$?
Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
- 125
1
vote
0
answers
61
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How to obtain value range of empirical PDF, given the mode and area? [closed]
Given an empirical PDF of a continuous random variable $X$, then integrating over its entire defined domain will yield an area of size 1. To find the probability of $X \ge x_1 \land X \le x_2 $ (as in ...
- 163
3
votes
1
answer
3k
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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 ...
- 153
1
vote
1
answer
49
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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 ...
- 13
7
votes
1
answer
267
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Integrating the inverse-Wishart density
It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
- 11k
2
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1
answer
290
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Expectation of $h \circ X$
I'm only starting to learn statistics.
The definition I've been given for the expected value (expectation) of a continuous random variable X with probability density function (PDF) $f_X$ is the ...
4
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2
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272
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Calculating multivariate integrals between lower and upper bounds
Suppose $\vec{X}=(x_1,x_2,...,x_n)$ follows some continuous multivariate distribution, such that $x_i\in{\rm I\!R}, i=1,...,n$.
Suppose also that I have access to the following functions:
$\phi(\...
- 268
1
vote
0
answers
68
views
Conditional expectation of the probability that a prior parameter is greater than some value
I am working in a Bayesian setting where I have a prior $p \sim \text{Beta}(\alpha, \beta)$. For reasons that don't really matter, I'm later defining a new parameter, call it $C$, which is the ...
- 163
2
votes
1
answer
295
views
Solving a marginalization integral involving exponential distributions
I'm trying to solve a marginalization integral
\begin{equation}
\int p(y,w) dw
\end{equation}
in order to compute the density $p(y)$.
I assumed the following model:
\begin{equation}
y = (u+w)^2 + v
\...
- 324
3
votes
1
answer
85
views
Why this integral converges to the beta distribution?
I know the beta distribution has a domain of [0,1] and I know the pdf but I just don't understand how the second last step here led to the last step.
- 73
6
votes
2
answers
2k
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What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?
Overall question: Assume X$_i$ ~ i.i.d. Unif(0,1). What is the expected length of a sequence that is monotonically increasing when drawn from the distribution?
The skeleton of the solution is
$$
\...
- 1,554
4
votes
3
answers
259
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pdf of $Z_{(n)}/Z_{(1)}$, where $Z \sim \text{Exponential}(1)$, i.i.d. (order statistics)
Let $Z_i \sim \text{Exponential}(1)$, iid, $i=1,2, ..., n$ and let $Z_{(i)}$ be the $i^{\text{th}}$ order statistic.
How can I find the pdf of $T=Z_{(n)}/Z_{(1)}$?
(the ratio of maximum and minimum ...
- 467
2
votes
1
answer
1k
views
Expectation using CDF
I have a problem understanding the solution of an exercise:
$F_x(x) = 1-(\frac{\sqrt{3}}{2}x + \frac{3}{2})^{-3}$ for $x \geq \frac{-1}{3} \sqrt{3}$, $0$ elsewhere
and i am asked to compute the ...
- 73
2
votes
1
answer
196
views
The median of the absolute value of the difference of two dependent log normal random variables
Assume X and Y have a bivariate lognormal distribution (x,y>0) that is:
$f_{X,Y}(x,y)$=$$\frac{1}{2p\sqrt{1-r^2}xy\sigma_1\sigma_2}exp\{\frac{-1}{2(1-r^2)}[(\frac{ln(x)-\mu_1}{\sigma_1})^2-2r(\frac{...
- 23
9
votes
4
answers
1k
views
Integral identity of lemma contained in infoGAN paper
I've come across a lemma in the infoGAN paper. I do not understand the derivation of Lemma 5.1 in the addendum of the paper. It goes as follows (included as png):
I do not understand the last step. ...
- 670
0
votes
0
answers
201
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probability density function of a cumulative density function
I was reading that the probability density function (pdf say $f(x)$) of a cumulative distribution function (cdf say denoted as $F(x)$) is uniformly distributed from 0 to 1.
Basically this is what I ...
- 649
2
votes
1
answer
541
views
Integration of student's T PDF
The standard normal distribution has the property that
$$\int_{-\infty}^\infty \phi(x)\phi(x+a)dx = \frac{1}{\sqrt2}\phi\left(\frac{a}{\sqrt2}\right)$$
How would I go about proving that the same ...
- 556
1
vote
1
answer
96
views
An integration problem containing a P.D.F. and a score function
Given, a probability distribution function (P.D.F.), $f_{\theta}(x)$ and its score function, $$u_{\theta}(x) = \frac{\partial}{\partial \theta} log_e f_{\theta}(x) \;\;,$$
how do I evaluate the ...
- 579
0
votes
1
answer
82
views
How to show $\int_{-\infty}^{\infty}f_{X}\left(x\right)\left(P\left(X<x\right)-P\left(X>x\right)\right)dx =0$
For a continuous random variable X
, intuition tells me that $$\int_{-\infty}^{\infty}f_{X}\left(x\right)P\left(X<x\right)dx=\frac{1}{2}$$
and more weakly that $$\int_{-\infty}^{\infty}f_{X}\left(...
- 291
0
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0
answers
36
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Is the following integral of a pdf an identity, i.e. always true?
I am reading a paper and the author starts a proof with this
$$
p(\hat{R}|R) = \int p(\hat{R},\theta|R)d\theta
$$
p is the density function.
Is this something that is always true? Can you help me ...
- 1,275
5
votes
1
answer
983
views
Multivariate probability density - Help with an integral
I'd like some help solving this problem about multivariate probability densities.
Let the random variables X and Y have the joint density f(x,y) = 1/y
for 0 < x < y < 1 and 0 otherwise. ...
- 53
3
votes
1
answer
2k
views
Overlap between two normal pdfs [duplicate]
I have two normally distributed random variables (estimated from two different sets of samples), and I'd like to know how "similar" those variables are (in order to compare the sets).
I had the idea ...
- 171
27
votes
4
answers
31k
views
"The total area underneath a probability density function is 1" - relative to what?
Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%.
But I ...
- 974
2
votes
1
answer
86
views
Explanation of density rewriting?
Can somebody please explain the math behind this statement to me? I am not sure how they represent the left hand side by that integral and finally how it is proportional to that.
\begin{align}
p(S_{t+...
- 1,170
2
votes
2
answers
154
views
Ordered gamma variables led to an ugly integral
Suppose $X_1,X_2,...X_n$ are i. i. d. random variables with p. d. f.
$$f(x)=xe^{-x}I_{(0,\infty)}\!(x)$$
and let $Y_1,...,Y_n$ be the order statistics for these variables.
a) Find the conditional p. ...
- 425
3
votes
2
answers
256
views
What is the expected partial value function really called?
If f is a pdf, the integral of xf(x) over the entire range where f(x) > 0 gives, of course, the expected value. Suppose that integrate the same function, xf(x) from negative infinity up to t, ...
- 3,247
3
votes
1
answer
105
views
Find the pdf of Y
$$ f(x)=\frac{1}{2}e^{-|x|} , -\infty < x < \infty ;
Y=|X|^{3} \
$$
I understand that I have to divide it in two parts and write it in cdf form
$$
F_{x}(y^\frac{1}{3}) - F_{x}(-y^\frac{1}{3}) \...
- 851
7
votes
1
answer
470
views
Is output of Deamer deconvolution not a density?
I have a Model Y= X+e and need the density of X. The deamer package deconvolves the density for X, but if I use the simpsons rule to integrate this density, I get values which are above 1.
The ...
- 1,411