All Questions
Tagged with integral or integration
1,507 questions
1
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1
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Proving bound on expectation of likelihood ratio involving mixtures
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
- 13
4
votes
0
answers
69
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Integration of volume forms over manifolds with corners
Suppose that $M$ is a (compact, oriented, smooth) manifold with corners.
Let $M_{\leq 1}$ the manifold with boundary of all points of index $\leq 1$ (following the notations here). This manifold is an ...
- 1,035
0
votes
1
answer
157
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Weak convergence of $f(x,e^{itx})$
This is the desired result (what I want to prove):
$$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$
Given that $f\in C([a,b]\times\{e^{i\...
- 233
2
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0
answers
101
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An inequality related to Problem 10210 AMM 1992 No. 3
Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
13
votes
2
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483
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Closed form of $\frac{1}{\pi^{n}}\int_{\mathbb{R}^n}dV \prod_{i=1}^{l}\frac{1}{1+(v_{i}^{T}x)^2}$
Is it possible to find closed form of $$I=\frac{1}{\pi^{n}}\int_{\mathbb{R}^n}dV \prod_{i=1}^{l}\frac{1}{1+(v_{i}^{T}x)^2}$$
in terms of vectors $v_i$?
Where $x=(x_1,\ldots,x_{n}),\ dV=dx_1\wedge\...
- 233
1
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0
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146
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integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 501
0
votes
1
answer
115
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Fourier transform of exponential over torus
I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
0
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0
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36
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Contribution of Fisher information near jump points in convolved probability distributions
I am trying to compute the contribution to the Fisher information from jump points $b_i(\theta)$ in the convolved function $f(x; \theta)$ with respect to the parameter $\theta$. I am unsure whether it ...
- 99
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0
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42
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How can I solve this non-local optimization problem?
I would like to find a continuous function $u:[0,1]\to \mathbb{R}$, which is a minimizer of the following functional
$$ F(u) = \frac{1}{2}\int_0^1 \int_0^1 \left( \frac{u(x)-u(y)}{x-y}\right)^2\mathrm{...
0
votes
2
answers
148
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Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 21
1
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2
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117
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If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
4
votes
1
answer
140
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Integrate unit normal vector over unit sphere intersected with a simplicial cone
Let $S^{d-1}$ be the unit sphere in $\mathbb R^d$. Consider a ($d$-dimensional) simplicial cone $C$ in $\mathbb R^d$ whose extremal rays are spanned by some unit vectors $\mathbf{u}_1,\ldots,\mathbf{u}...
- 135
2
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2
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382
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Definition of Multivariable Antiderivatives
In the 1-dimensional case antiderivatives $F(x)$ of a function $f(x)$ have the following properties:
$F(x)=\int\limits_0^xf(t)dt$
$\frac{d}{dx}F(x)=f(x)$
$\int\limits_a^bf(t)dt = F(b)-F(a)$
Of ...
- 13.2k
1
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1
answer
189
views
How to evaluate the following integral?
How to (analytically) calculate the following integral,
$$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$
where $\langle z, \zeta \...
- 501
-2
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0
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113
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How to calculate this integral [migrated]
Is there a formula of this integral
$$\int_{S_{2n-1}} \frac{e^{i a \langle z, \zeta \rangle}}{|z - \zeta|^{2\lambda}} \, d\sigma(\zeta)$$
and how to calculate it.
Thank you in advance
- 113
1
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1
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141
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Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function
Define a function $f \in C^m[-1,1],m \in \mathbb{N}$ and $g\in C^{\infty}[-1,1]$. Also define a quadrature rule $Q$ for approximating the integral $\int_{-1}^1 h(x)dx$ for some function integrable ...
- 69
1
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1
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112
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Bounding a Riemann sum by its integral limit?
Let $M_{n}(\mathbb{C})$ denote the space of complex $n \times n$ matrices and, for $a>0$, $a \in \mathbb{R}$ fixed, let $A: [0,a) \to M_{n}(\mathbb{C})$ be a given function. I will write $A(t) = (...
1
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1
answer
62
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MGF relevant to modified 2nd kind Bessel
Given the moment-generating function
$$
m_{0}(t)=\frac{1}{\sqrt{1-t^2}}\,\text{ for }t<1,
$$ which corresponds to a distribution with density
$$
f(u) = \frac{1}{\pi}K_{0}(\frac{u}{\pi })
$$ where $...
0
votes
1
answer
97
views
Numerically bounding a Exponential-Trigonometric Integral [closed]
I am having some trouble with this (undergrad) problem. The Twitter account I found this from was deleted so I unfortunately have not found an answer.
I have tried decomposing into Riemann sum and ...
- 13
6
votes
1
answer
853
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Is integration semi-algebraic?
I am learning a bit of semi-algebraic geometry and I have looked into a bunch of examples of functions which are semi-algebraic. In particular I have tried to understand whether the function
$$ F: (0,\...
- 1,045
5
votes
0
answers
160
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Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
4
votes
1
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228
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A definite integral of a hypergeometric series related to the enumeration of fusenes
If my calculations are correct, the number of (not necessarily planar) fusenes of perimeter $2n$ grows asymptotically like $\mu^n$ where
\begin{equation}
\mu = \frac{6}{\int_0^{1/6} H(t)\mathrm{d}t} = ...
- 3,927
9
votes
1
answer
429
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A curious norm related to the L¹ norm
If $f \in C^0([0,1])$, one can define
$$\Vert f \Vert_? = \sup_{J \subset [0,1]} \left\lvert \int_J f \right\rvert,$$
where $J$ runs among all subintervals of $[0,1]$.
This is a norm on $C^0([0,1])$ (...
- 575
2
votes
0
answers
104
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Existence of Dirac measures in the context of joint and marginal distributions
Let $\pi$ be the joint law of $(X, Y)$ with marginal distributions $\mu$ and $\nu$. We assume that we have: for all $A \in \mathcal{B}(\mathbb{R})$ such that $\mu(A) > 0$
$$
\nu\left(\{y \in \...
2
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0
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75
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Have the open Questions 1 and 2 from Section 7 of the paper "Integrals with values in Banach spaces" been answered?
Some context: I had previously asked the post below on MSE, but someone suggested I ask it here and delete the original post.
In section 7 of the paper Integrals with values in Banach Spaces and ...
- 121
3
votes
2
answers
336
views
An Integral invoving products of modified bessel functions
I am a physicist working on a problem where the following integrals are concerned:
$$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$
$$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,...
- 41
3
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1
answer
242
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Evaluating $\int_0^{10} \operatorname{sn}(x\mid i) \, dx$
I am trying to evaluate:
$$\int_0^{10} \operatorname{sn}(x\mid i) \, dx$$
where $\operatorname{sn}$ denotes the Jacobi elliptic function sn.
The indefinite integral is:
$$(-1)^{3/4} \tanh ^{-1}\left(\...
2
votes
0
answers
99
views
Closed form for $\int_0^{+\infty} \ln^p(t) \frac{\sin^q(t)}{t^r}dt$
Do you know if there exists a closed form for the integral :
$$I_{p,q,r} = \int_0^{+ \infty} \ln^p(t) \frac{ \sin^q (t)}{t^r} dt$$
where $p$, $q$, $r$ are natural integers such as this integral ...
- 69
4
votes
1
answer
214
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Explicit expression for a function in number theory
In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of ...
- 1,247
2
votes
1
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315
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Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?
When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$
I_k=\int_{0}^{\...
- 1,091
3
votes
1
answer
380
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Application of Feynman parameters in an improper integral
I've read in a paper
E. Guadagnini, M. Martellini, M. Mintchev, Wilson Lines in Chern-Simons Theory and Link Invariants, Nucl. Phys. B 330 (1990) pp 575–607 https://doi.org/10.1016/0550-3213(90)90124-...
- 149
4
votes
1
answer
257
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Asymptotics of an entire function with real zeroes on the real line
Define $ F(x) := A x^m e^{B x} \prod_{k \geqslant 1} (1 - x/\alpha_k)e^{x/\alpha_k} $ where we suppose that $ \alpha_k \in \mathbb{R} $. This function is defined on the whole complex plane and is ...
- 593
3
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1
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79
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Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$
Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
- 407
0
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1
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87
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Properties of slowly-varying functions at two large points
Consider a slowly-varying function $L:(1,\infty) \mapsto (0,\infty)$, i.e. a function such that $L(cx)/L(x)\to1$ as $x\to\infty$ for all $c>0$. Assume that $\lim_{x \to \infty}L(x)=0$.
Question: is ...
- 195
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0
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75
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$\int_{-\infty}^\infty (\log(\cosh(y))^k e^{-(y-\mu)^2/2\sigma^2} \mathrm{d}y$
Given $\mu \ge 0$, $\sigma > 0$, and $k \in \mathbb{N}$, I want to compute $$\frac{1}{\sqrt{2\pi\sigma^2}} \int_{-\infty}^\infty (\log(\cosh(y)))^k e^{-(y-\mu)^2/2\sigma^2} \mathrm{d}y = \frac{1}{\...
- 141
9
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2
answers
2k
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Why does this theta function value yield such a good Riemann sum approximation?
Let $T(q)$ denote the third Jacobi theta function at $z=0$; i.e.,
$$T(q) := \sum_{n\in\mathbb{Z}} q^{n^2}.$$
Then for any $x>0$, $T(\exp(-1/x)) \approx \sqrt{\pi x}$. For example, as the entry for ...
- 82.7k
0
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1
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128
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Characterizing the integral as a function of $n$
Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
- 25
0
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1
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91
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Integral of complementary error function times exponential with polynomial argument
I try to understand the behavior of the following integral as $a\rightarrow\infty$
$$I(\delta)=\int_{-\delta}^\delta{\rm erfc}\Big(-\frac t{\sqrt a}\Big){\rm e}^{bt-t^2/a}{\rm d} t,$$
where ${\rm erfc(...
- 139
0
votes
0
answers
42
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Is this function $\mathcal{C}^1$ in the global sense?
Denote by $\mathbb{U}$ the complex unit disk. Let $\mathcal{O}$ an nonempty open subset of $\mathbb{R}^n$ $(n\geq 1)$, and $f\in\mathcal{C}^1(\mathcal{O}\times\mathbb{R},\mathbb{U})$ such that for all ...
- 449
1
vote
0
answers
35
views
Is it possible to manipulate heat kernel on H-type groups?
In Nathaniel Eldredge's work see here, he uses the explicit expression of the Heat Kernel on H-type groups (for example the Heisenberg group is an H-type group):
$$p_t(x,z)= (2\pi )^{-m} (4 \pi )^{-n}...
- 677
24
votes
1
answer
1k
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Integrating on $\mathbb{R}$ by summing on $\mathbb{Q}^+$
Does the following integration method hold for regular enough functions $F:\mathbb{R}\to\mathbb{R}$?
\begin{align}
&\zeta(2)\sum_{\frac{a}{b}\in\mathbb{Q}_n} \frac{F(\log \frac{a}{b})}{\sqrt{abn}...
- 634
0
votes
2
answers
116
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Help on integral regarding analytical Fourier transform
To explain my problem I start with two functions to be sine transformed. This question is a problem of current research in the field of electrolyte transport theory. The first Function is given by:
$$...
1
vote
0
answers
81
views
An integral containing modified Bessel functions
During my studies I am facing the following problem. Let $I_\nu(x)$ be the modified Bessel function for $\nu\in(0,1/2]$.
I want to compute the following integral (it is are resolvent)
$$
R(z) = \frac{...
- 11
5
votes
1
answer
355
views
Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty$
I want to know whether or not
$$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}} \frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty.$$
Here $D $ denotes the ...
- 61
6
votes
0
answers
150
views
Can this Casimir-effect integral be reduced to a special function?
This integral plays a central role in a physics problem (Casimir effect)${}^\ast$
$$\Omega(\phi,L)=-\frac{1}{\pi}\operatorname{Re}\int_0^\infty \ln\bigl[1+\beta(\omega)^2 e^{i\phi-2\omega L}\bigr]\,d\...
- 188k
0
votes
0
answers
188
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Why is the property of linearity against an infinitely-large factor considered essential for surreal integration?
Why the property $(b)$ in Proposition 14 in this paper on surreal integration is considered essential?
The Proposition lists the desired properties of the surreal integration, and among others lists ...
- 10.1k
6
votes
1
answer
501
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In surreal numbers, what are the main difficulties so far in defining integration?
I know, there were several (including unsuccessful) attempts at defining integration on surreal numbers, so I am asking for a good summary of what have been the main difficulties so far.
Particularly,
...
- 10.1k
1
vote
1
answer
62
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Integrability in the product space can follow from a property of the Nemytskii operator?
Let's say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where ...
- 1,759
0
votes
0
answers
115
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Integral of a measurable function with parameter is measurable?
Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that:
$f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$
$f(\...
- 1,759
0
votes
0
answers
149
views
Reference book for a probability course
In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
- 1,561