Questions tagged [density-function]
For questions on using, finding, or otherwise relating to probability density functions (PDFs)
2,132
questions
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Finding the CDF and the PDF of random variable $ Y=C\sqrt{r^2−B^2}^D$
For a random variable given as $R$ with pdf $$f_R(r) = \frac{12}{73} \left(\frac{27r}{M^2}-\frac{35r^3}{M^4}+\frac{8r^5}{M^6}\right)$$
I need to find pdf and cdf $$ Y=C(\sqrt{r^2−B^2})^D$$
where $C , ...
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1
answer
59
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Does every density have random variable.
I am studying probability, everywhere i read, the following is stated as a fact: if $f$ is measurable and positive, and $$\int_{\mathbb{R}}f(x)dx = 1$$ then $f$ is density of some random variable. Is ...
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NN Taleb's A Short Note on P-Value Hacking, what is this standard result for a transformation?
I am reading Taleb's note on p-value hacking (https://arxiv.org/pdf/1603.07532.pdf) and am pretty confused with Proposition 1.
Let $Z$ be a random normalized variable with realizations $\zeta$, from a ...
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1
answer
54
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Solving an antiderivative via u-substitution
I have a density function defined as $f(x) = (1/10) \exp(-x/10)$. The answer to the solution suggests that the antiderivative, the cumulative distribution function, ought to be $\exp(-x/10)$, however ...
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1
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53
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What is the distribution of distances between points defined by a multivariate Gaussian? [closed]
Assume points x and y in N dimensional Euclidean space (1-7 dimension for my application), with dimension values xj, yj, j=1:N.
The locations of x and y are described by multivariate normal ...
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votes
1
answer
76
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How to find CDF from PDF for jointly random variables?
Any help with this question?
Let A be the set in $R^2$ bounded between the lines $y = 0$, $x = y$ and $x = 2$.
Let $(X, Y)$ be jointly continuous random variables with pdf $f(x, y) = cxy^2$ for $(x, y)...
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32
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Applications: Cauchy-Schwarz inequality and Triangle inequality
I am reading a proof from "Transporting Probability Measures" (PhD's thesis of Gilles Mordant) with the following steps that I am not sure to understand.
Let $X \sim P$ where $P$ is a ...
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1
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1
answer
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Hartree equation in density matrix formalism is equivalent to a system of Hartree equations
I want to prove that:
If the density matrix $$\gamma(t):=\sum_{j=1}^N|u_j(t)\rangle\langle u_j(t)|=\sum_{j=1}^N\langle u_j(t),\cdot\rangle u_j$$ solves $$i\partial_t\gamma(t)=[-\Delta+w*\rho,\gamma],$...
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2
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Why must the population mean equal the population median if the density function is symmetric about the median?
Suppose we have $X_1,X_2,...X_n\overset{\text{iid}}{\sim}f(x-\mu)$ where, $f$ is symmetric about $0$. So, $\mu=$ median of $X$ and (if it exists) mean of $X$.
Since, $f(x)$ is symmetric around $0$, $...
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2
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36
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Papa Rudin $4.16$ Lemma [closed]
There it is :
Suppose that
$(a)$ X and Y are metric spaces, $X$ is complete,
$(b)$ $f$ : $X$ $\to$ $Y$ is continuous
$(c)$ $X$ has a dense subset $X_0$ on which $f$ is an isometry, and
$(d)$ $f(X_0)$ ...
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1
answer
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Solving the variance of a compound pdf
Before I state the question, I want to quickly point out that the question is stated using a constant $\sigma$, while also maintaining that the inner pdf $f$ is a mean 0 and variance 1 density. It's ...
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2
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Finding the mean from a probability density function
This one is worded a little weirdly, since the question I found seems to maintain that there exists an unknown mean for the PDF and a "mystery number" represented by $\mu$. Just something to ...
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1
answer
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Finding distribution and density function of $X^2/(X^2+Y^2)$ where $X,Y∼N(0,1)$
I have two random independent standard normal variables $X,Y∼N(0,1)$.
How can I find the distribution of $\,\dfrac{X^2}{X^2+Y^2}\;?$
I know that if we talk about only $X^2$ then it will be a Chi-...
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votes
2
answers
43
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Find the conditional density of $X_1$ given that it is not the smallest of the n values among n independent uniform $(0, 1)$ random variables
Let $X_1,...,X_n$ be independent uniform $(0, 1)$ random variables. Find the
conditional density of $X_1$ given that it is not the smallest of the $n$ values.
Here is my idea:
Let $Y$ denote that $X_1$...
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1
vote
0
answers
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Proving that the bias of the derivative of Parzen-Rosenblatt (kernel density) estimator is of order $O(h^2) $ and $O(h)$ when $h$ approaches $0$
I'm trying to calculate the bias of this estimator of $f$ a $C^4$ mesurable function:
$$\hat{f'}_{h,n} = \cfrac{1}{nh^2}\sum_{j=1}^n K'\left(\cfrac{x-X_j}{h}\right) =\cfrac{1}{h^2}K'\left(\cfrac{x-X_1}...
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Finding the mean value of a Pareto Distribution.
The question:
Quality control experts estimate that the time (in years) until a specific electronic part from an assembly line fails follows a Pareto density of the form $f(x) = 3/x^{4}$ for $1 < x ...
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limit or maximum value of a bivariate function
the expression below is a function of two variables, $x$ and $y$, with $y>x>0$ and both being standard normally distributed, where $\phi$ is the density of the standard normal $N(0,1)$ and $\Phi$...
1
vote
0
answers
55
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Probability of a random variable being greater than the maximum among others plus a constant
Suppose we have $N+1$ i.i.d. random variables from a normal distribution with mean $0$ and variance $\sigma^{2}$. That is:
$$X_{i}\sim \mathcal{N}(0,\sigma^{2}),\quad i=1,\dots , N+1$$
Suppose $b>0$...
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1
answer
65
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The density of a pushforward probability measure: the reciprocal of the Jacobian determinant?
$
\def\dee{\mathop{\mathrm{d}\!}}
\def\Jac#1{\mathop{\mathbf{J}_{#1}}}
$
I'm confused about how to use the change of variable formula to describe the density of a pushforward measure. My question ...
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1
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1
answer
34
views
Finding the median value from a PDF
Suppose that the time in days until hospital discharge for a certain patient population follows a density $f(x) = (0.5)\exp(-x/2)$ for $x > 0$. What is the median discharge time in days?
I reason ...
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2
votes
2
answers
163
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Calculating $\mathbb{E}[2\sin (\pi Z)|\cos (\pi Z)]$ when $Z$ is uniform on $[0,2]$
I am trying to calculate the following conditional expectation.
Let Z be a uniformly distributed random variable on the closed interval $[0, 2]$.
Define $X = \cos(\pi Z)$ and $Y = 2\sin(\pi Z)$.
...
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votes
1
answer
50
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A pdf in R is positive definite iff the real part of its characteristic function is nonnegative?
For a continuous bounded pdf $K(x)$ that is symmetric about $0$ (also suppose $K(0)>0$), let $\phi(t):=\int e^{itx}K(x)dx$ be its characteristic function. Does it hold that $K(x)$ is positive ...
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2
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56
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symmetric function derived from a symmetric continuous pdf is positive definite?
Let $K(x)$ be a continuous bounded pdf symmetric about $0$; $K(x)$ is increasing for $x<0$ and decreasing for $x>0$. If $k(x,y):=K(x-y)$, then is $k(x,y)$ a positive definite function?
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Solving for the quantile of a probability density function.
What is the quantile, p, from the density $e^{-x}(1+e^{-x})^{-2}$?
I believe I am on the right path to the solution, but I am stuck part way through.
I figure that the CDF is almost certainly $(1+e^{-...
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2
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63
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What is the codomain of the Radon–Nikodym derivative and why$
My question is regarding the Radon–Nikodym derivative $\frac{d\nu}{d\mu}$, when $\nu \ll \mu$ and both measures are $\sigma$-finite. So Wikipedia says in the article about the Radon–Nikodym theorem ...
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Show: If a density exists, then the involved measures are absolutely continuous
Let $(X, \mathcal{A})$ be a measure space. We can define a measure $\nu$ w.r.t. to another measure $\mu$ and a function $f:X\rightarrow \bar{\mathbb{R}}^+$ (called the density) as follows:
$$\nu(A)=\...
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1
answer
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Why am I computing the PDF of this modification of a normal distribution wrongly?
Let $Z$ be a standard normal variable (mean $0$ and variance $1$), $\theta\in \mathbb{R}$ and $X=\theta+|Z|$. I want to find the density of $X$.
I began by computing the cummulative function. So, for $...
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answers
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Calculate probability density function of a random vector $(X + Z, Y + W)$
Let $(X, Y)$ and $(Z, W)$ be independent random vectors with the same density:
$$f_{X,Y}(x,y) = e^{-x}\frac{1}{\sqrt{2\pi x}}e^{-\frac{(y-x)^2}{2x}}$$
for $x > 0$ and $y \in \mathbb{R}$. Find the ...
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1
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37
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Probability density of $X_1 − 2X_2$, where $X_1, X_2$ independent and exponentially distributed.
Problem
Consider $X_1, X_2 \sim \varepsilon(1)$ independent and define $T = X_1 - 2X_2$.
I want to calculate the probability density function (pdf) of $T$, denoted by $f_T$, which can be obtained by ...
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0
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29
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Probability Density Function Irregularity
I'm currently working on a question which asks me to solve for the density function of a function of the random variable $Y$; $U = (Y-2)^2$. The given density function of $Y$ is as follows:
$
\ \begin{...
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1
answer
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Understanding $P(XY\leq c)$
Let be $(\mathbb{R},\mathcal{B}(\mathbb{R}),P)$ a probability space and $X,Y$ two real valued absolutely continuous random variables, where $f_X$, $f_Y$ and $f_{XY}$ are the respective density ...
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1
vote
1
answer
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Compute the density of $XY$
Let be $X,Y$ two independent real valued random variables, with density functions $f_X$ and $f_Y$. Compute the density function of the product $XY$, where $X\geq 0$.
My approach:
First, I think that ...
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0
votes
1
answer
81
views
Variance of a location-scale transformed density
NOTE: I originally wrote this question by trying to transcribe the original problem according to my understanding, and in the process I introduced some problems. I used 'asterisks' as a ...
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1
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1
answer
59
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PDF of $|X-Y|$ when X and Y are independent uniform on $\left[0,l\right]$
pretty much trying to solve the question in the title, what I tried is:
consider $Z=\left|Y-X\right|$, and try to compute $F_{Z}\left(t\right)=1-P\left(Z>t\right)=1-\left(P\left(X-Y>t\right)+P\...
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2
votes
0
answers
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Non-existence of Lebesgue probability densities
A standard result taught in mathematical statistics courses is that the multivariate gaussian only has a density if the covariance matrix is non singular: i.e. if $X \sim N_p(\mu, \Sigma)$ and $\Sigma$...
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0
answers
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Conditional density of ordered iid Normal random variables
I am trying to apply the change of variable theorem for the calculation of the distribution, via the conditional density, of this transformation of joint iid $X_i$ that are N(0,1) variables with no ...
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0
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21
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Compute conditional probability that may not have a joint density
Using the measure-theoretic definition of conditional expectation, one can show that if random vectors $X\in R^m$ and $Y\in R^n$ have a joint density with respect to the Lebesgue measure in $R^{m+n}$, ...
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1
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56
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Can the probability density be defined variationally?
Define a random variable $X \in \mathbb{R}$. Suppose that the distribution of $X$ is dominated by the Lebesgue measure and hence admits a density (pdf) $f^*(x)$ at each $x \in \mathbb{R}$.
Is it ...
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3
votes
1
answer
118
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Conditional density of ordered iid Exp(1)
I am trying to apply the change of variable theorem for the calculation of the distribution, via the conditional density, of this transformation of joint iid $X_i$ that are Exp(1) variables with no ...
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3
votes
1
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155
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Statistical guarantee on the effectiveness of a kernel density estimation
I have recently started studying about kernel density estimations. For reference, if one is given if $(x_1,x_2,...,x_n)$ is an i. i. d. sample drawn from an (unknown) distribution with an unknown $f$ ...
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3
votes
1
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Ordered iid Exp(1) Random Variable: Conditional Density Function
I'm working on the following exercise from Achim Klenke's "Probability Theory: A Comprehensive Course" (3rd Ed, Exercise 15.1.3):
Let $n \in \mathbb N$ and let $X_1, \ldots, X_n$ be i.i.d. ...
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0
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50
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Joint probability mass function, definition problem
Let $p_1, p_2 \in (0, 1)$ arbitrarily fixed. Consider the following discrete joint density:
$$p_{X_1,X_2}(k,k) = p_1(1 − p_2)^kp_2 \text{ for } k ≥ 0 \text{ integer };$$
$$p_{X_1,X_2}(h,h-1) = (1 − ...
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vote
1
answer
36
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Proof by density Sobolev Spaces?
When reading about Sobolev Spaces, some results are proven using the density of smooth function.
So they prove the results on smooth function and then conclude by taking limits within the integral.
...
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2
votes
1
answer
45
views
probability density function of $X^2$ and $e^X$ based on $X$
Let $X$ be a random variable with probability density function $f_{X} \left( t \right)$, which for every $t<0$ satisfies: $f_{X} \left( t \right)=0$, and for every $t \geq 0$ satisfies: $f_{X} \...
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0
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12
views
Computing conditional PDF of continuous R.V.s
The problem is explained as such:
A soccer player is shooting at a round target.
The player chooses the radius of the target, notated $R$ from an exponential distribution $R$ ~$Exp(2\lambda )$.
...
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votes
0
answers
25
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How to deduce the joint distribution of two different elliptically contoured distribution. See details in following.
if $X \sim EC(0,\Sigma_1, g_1)$ and $Y\sim EC(0,\Sigma_2,g_2)$, then what is their joint distribution？That is, let $Z=(X,Y)^T$, what might the distribution of $Z$ be? Clearly, when $X$ and $Y$ are ...
0
votes
1
answer
28
views
Conditional PDF notation when conditioning on variable falling within certain range
I understand that the conditional PDF of some RV $X$, given some variable $Y$, is written as $f_{X|Y}(x|y)$. Also, I understand that if I want to describe general features of that function, I am to ...
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2
answers
60
views
Finding the PDF of a sum of 2 different distributions
Let $X,Y$ be two continuous independent R.Vs such that $X $~ $Uni(-1,3)$ and $Y$ is a R.V. defined within $[0,1]$ with PDF: $f_Y(y)=\frac {1}{\sqrt {y}}$, Find the PDF of $Z=X+Y$
In my first attempt, ...
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0
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2
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37
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Probability: Finding the probability density function when we have the expectation
I'm attempting to solve this problem, and I'm not sure how to (sort of) backtrack the PDF when I have the expectation.
Let $Y$ be a continuous R.V. given by the PDF: $f_Y(y)=2(1-y)$ where $y\in [0,1]$
...
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3
votes
0
answers
47
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Power law dependence in the distribution of $x = GPF(n) / \sqrt{n}$, where $GPF(n)$ is the greatest prime factor function
Consider the set of integers between $2$ and some large number $N$. For each such number $n$, we can compute the quantity $x = GPF(n)/\sqrt{n}$, where $GPF(n)$ is the greatest prime factor function. $...
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