Questions tagged [density-function]
For questions on using, finding, or otherwise relating to probability density functions (PDFs)
2,331 questions
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Gaussian tails for the density of the solution to an SDE driven by a fractional Brownian motion
Let $B^H$ be a one dimensional fractional Brownian motion with Hurst parameter $H\in(0,1/2)$ defined over a complete filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})...
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Is it possible to draw 2d density plot only knowing totals in any cross line
So we see from the density plot that the most dense part happens to be intersection of the x, y histograms where the highest bar meet. But suppose we only have the x, y histograms, I think it's not ...
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Why is the mean of a piecewise probability density function the sum of the mean of both separate intervals?
I am doing probability density functions for Calculus 2 and came across a problem where I had to find the mean for a piecewise function. I looked up how to find the mean in this case and the equation ...
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Pointwise limit of a sequence of density functions
I am trying to prove that the following sequence of density functions $$f_{n}(x) = \mathbb{1}_{\left(0,2^{n}\right)}(x) \cdot \left(\dfrac{\left(n\ln(2) - \ln(x)\right)^{n}}{2^{n}\cdot n!} \right)$$
...
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confusion about densities of a recursively defined sequence of uniform random variables
This is a follow up question to this question in which the following was asked:
Let $\left\{X_{n}\right\}_{n\in \mathbb{N}}$ be random variables such that $X_{0} \sim \text{Unif} \left(0,1\right)$ ...
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Expected value of a transformed absolutely continuous random variable
I am trying to prove this theorem:
Theorem 1. Given $X$ an absolutely continuous random variable with density $f_X$ and $Y=g(X)$ a transformation, we have that the expected value E[Y] of $Y$ is ...
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Can a probability density function not integrate to 1 over its support?
I don’t know much Probability Theory beyond the undergraduate level.
I was trying to model a simple scenario with my family. What is the probability I will develop type 1 diabetes in the following ...
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Finding a density function for the minimum of two uniformly distributed random variables. [duplicate]
Problem:
Let $Y_1$ and $Y_2$ be independent and uniformly distributed over the interval $(0,1)$. Find the
probability density function for the following:
$$ U = \min\left(Y_1,Y_2\right) $$
Answer:
Let ...
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How to derive the density formula for $Y=h(X)$ when $h$ is not injective?
Let $X$ be a real-valued random variable with density $f_X$, and let $Y = h(X)$, where $h: \mathbb{R} \to \mathbb{R}$ is a continuously differentiable function. Assume that for every $y \in \mathbb{R}$...
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Derivative of function of probability density function
Let $f(p)=p \log p$, where $p$ is a probability density. Now the task is to compute the derivative of this expression. I know that the solution is $f'(p)=\log p +1$. And when using standard rules from ...
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Characterize $d\in\{0,1\}^n$ for which $\sum_{j=0}^{n-1}d_j\binom{n}{j}>2^{n-1}-1$
Update: I've reduced the problem to characterizing $d\in\{0,1\}^n$ for which $\sum_{j=0}^{n-1} d_j\binom{n}{j} > 2^{n-1}-1$.
I’m interested in computing the volume of $$P_n := \left\{x\in\mathbb{R}...
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Deriving the density of spherical distribution from its characteristic function
It is well known that the characteristic function $\varphi(\mathbf{t})$ of $\mathbf{x}\in\mathbb{R}^{n}$ following a spherical distribution is of form
$$
\varphi(\mathbf{t})=\phi(\mathbf{t}^{\mathrm{...
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Why for $f$ here it must be for $x, y \in [0, + \infty)$ then $f(x + y) \le f(x) + f(y)$
I am reading in" A. Aizpuru, Density by moduli and statistical convergence, Quaest. Math.
37(4) (2014), 525–530."
Its link
In page 529:
Lemma $3.4.$ If $A \subset \mathbb N$ is infinite ...
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Probability density functions of discrete distributions?
If I understand it correctly, a discrete probability distribution is not absolutely continuous with respect to the Lebesgue measure because a set with zero Lebesgue measure can have a positive ...
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How to deal with such integration
I am working on a problem wherein i have to find outage probability. It is given as
\begin{multline}
P_1^o=\int_{z=0}^{\infty}\int_{h=0}^{\infty}\int_{y=0}^{\infty}\frac{1}{\Gamma(m_0)} \gamma\...
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Density probability for a discrete-time Markov Process
Imagine a Markov process $X_t\in[\underline{x}, \overline{x}]\subseteq [0,1]$, $t$ is discrete and $x$ is continuous. Introduce two continuous and differentiable functions $f(x)$ and $g(x)$, both ...
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Claryfing a passage in the calculation of the density of the Dirichlet distribution.
I'm trying to derive the joint density for the transformation:
$$
(x_1, x_2, \dots, x_{n+1}) \to \left(p_1, p_2, \dots, p_n, \sum_{j=1}^{n+1} x_j\right),
$$
where $p_i = \frac{x_i}{\sum_{j=1}^{n+1} ...
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1
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Bound on variance for logconcave density
I have been stuck with the following problem for a while, maybe it will come as obvious to some of you.
It states that there is some constant $C>0$ such that for any continuous logconcave density $...
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Unique solution to an equation involving convolution and probability densities
I am confronted with the following problem. Let $\overline{\mu}_s:=-\mu(x)1_{(-\infty,0)}(x)+\overline{\mu}(x)1_{(0,\infty)}(x)$ where $\mu$ is some cumulative distribution function and $\overline{\mu}...
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Proving $\nu = f\mu$ with $0 \leq f \leq 1$ when $\nu ≤ \mu$ in a σ-finite measure space
Let $(S, \Sigma, \mu)$ be a $\sigma$-finite measure space and let $\nu$ be another measure on $(S, \Sigma)$. Show that if $\nu(A) \leq \mu(A)$, for all $A \in \Sigma$, then $\nu = f \mu$ for some (a.e....
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Calculus of Variation with density functions
Suppose I want to maximize the following the functional:
\begin{equation*}
J[f] = \int_{0}^{1} m(x)f(x) \,dx
\end{equation*}
subject to the constraint that the density $f(x)$ integrates to $1$
\...
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1
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Density result in $ L^2(a,b;H_0^1(\Omega)) \bigcap C([a,b];L^2(\Omega))$
I want to prove the following proposition, but I face some gaps.
Proposition $\ \ \Omega$ is a bounded open set in $\mathbb{R^n}.$
Let $u \in$ $L^{2}((a,b);H_{0}^{1}(\Omega))\cap C([a,b];L^{2}(\...
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1
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Calculating $\mathbb E[Y|X]$ without calculating the marginal $f_{X}(x)$
This question was asked in our test but I am not able to solve it
Let $Y$ be Gamma random variable with parameters $n,\lambda$ (with $\lambda>0$ and $n$ a positive integer). That is, $Y$ has ...
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Prove that $n \int_{\mathbb{R}}\left[\cos\left(\frac{\xi}{nx}\right)-1\right]f(x)\,dx \to-c|\xi|$
Problem. Suppose that $f(x)$ is a continuous density of some random variable and support on $[-1,1]$, that is, $f(x)\ge0$ and $\int_{-1}^{1}f(x)\,dx =1$.
If in addition, $f(0)>0$ , I want to show ...
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Summation of any random variable and a gaussian random variable has a density function
I would like confirm if the below fact is true ask for a reference if it is true:
Let $X$ be a random variable, which does not necessarily have a density function. Let $G_{\sigma}$ be a gaussian ...
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