Does $E[E[X|X\le Y]]=E[X]$?
This is a question I came up with while doing some related calculations.Let $X$ be a random variable defined on some probability space $(\Omega, \mathcal A, P)$. To make everything as nicely behaved as...
View ArticleDefinition of improper integral, requirement or property?
I am confused about the definition of improper integral of more than one singularities, for instance, an improper integral of the form $ \int_{-\infty}^{\infty} f(x) \, dx $. From Wiki and some...
View ArticleLet S ⊂ R be a nonempty bounded set. Then there exist monotone sequences...
Let S ⊂ R be a nonempty bounded set. Then there exist monotone sequences ${x_n}$ and ${y_n}$ such that $x_n$, $y_n$ ∈ $S$ and $\sup S = \lim_{n→∞} x_n$ and $\inf S = \lim_{n→∞} y_n$How can I prove this...
View ArticleProve that this sequence is eventually one
Let $ \ \mathbb{N} = \{ 0,1,2,3,4,...\} \, $, $ \ O = \{ n \in \mathbb{N} : n \text{ is odd} \} \ $ and $ \ T: O \to O \ $ be such that, for all $ \ n \in \mathbb{N} \, $,\begin{align*}T(8n+1) & =...
View ArticleShow $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further,...
Let $f$ be integrable on $[a,b]$, and let Let $P = \{x_0,x_1,x_2,...,x_{n−1},x_n\}$ be any partition of $[a,b]$. Show that $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further,...
View ArticleHow to prove that such a n exists such that $f_n(x)=x+n$
This is a sample question from ISI-$2023$,let $f_n$ be a sequence of continuous functions for $n\ge 1$, suppose $\forall x \in \mathbb{R}$, $\exists n_x \in \mathbb{N}$, such that $f_{n_x}(x)=x+n_x$....
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How to see the solution of heat equation on whole line is uniformly Schwartz...
This is the result proved in Stein Fourier Analysis Chapter 5, 5.2 Corollary 2.2.Suppose $\Phi_{t}=\frac{1}{\sqrt[]{ 4\pi t }}\exp\left( -\frac{x^{2}}{4t} \right)$ is the heat kernal.It said that if...
View ArticleConfused on how the proof of Monotone Convergence Theorem proves convergence
Self learner here. This is the typical proof for the Monotone Convergence Theorem. Typically the pattern I've seen for proving convergence is based on the epsilon delta definition of a limit L of an...
View ArticleIs $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?
Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books...
View ArticleSigma-field of a sequence of Random Variables
Problem:Suppose $\tilde{X}=(X_1,X_2,\dots)$ is a sequence of RVs on $(\Omega,\mathcal{B})$. Prove that $\sigma(\tilde{X})$ is generated by events of the form:$$\bigcap_{i=1}^m \{X_i\leq x_i\}\mbox{ for...
View Articlegeneral form for effective magazine size with refunded bullets
This exploration started because of the video game destiny, in which weapons can have essentially magical perks, such as every 3rd shot hit, add one bullet back into the magazine.What I am curious...
View ArticleEvaluate Limit of Faulhaber Formula [duplicate]
Prove or evaluate that:$$ \lim_{ n \to \infty }\frac{S_{k}(n)}{n^{k+1}}=\frac{1}{k+1}$$Where$$S_{n}^k=\sum^{n}_{m=0}m^k$$So I have noticed that it can be done by proving $S_k(n)$ is a polynomial of...
View ArticleIf $f:[0,\infty)\to\mathbb{R}$ is a decreasing differentiable function, prove...
I am trying to solve the following problem:Let $f:[0,\infty)\to\mathbb{R}$ be a decreasing differentiable function such that $f(0)=1$, $\lim_{x\to\infty} f(x)=0$ and $\int_0^{\infty}x^4f'(x)dx <...
View ArticleRestricted version of the Kolmogorov–Arnold representation theorem
The Kolmogorov–Arnold representation theorem says that any multivariable function $f(\mathbf{x})$ where $\mathbf{x}\in\mathbb{R}^n$ can be written...
View ArticleShow that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}$ is convergent.
Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}} \,dx$ is convergent.The points $0$ and $1$ are the only points of infinite discontinuities of $\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}.$ The integral is...
View ArticleGiven a number of minutes, check if we should work on rest according to the...
I am trying to calculate if a specific minute is within a pomodoro (work interval) or if it's within a rest interval. Assuming I am taking breaks in the following intervals:Work for 25 minutes...
View ArticleDecomposition of a function
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function with compact support on $\mathbb{R}$.Under which hypotheses on $f$ the functional equation (in the $w$ variable)$$w(2t)-w(t)=f(t)$$admits at least a...
View ArticleDensity in $W^{1,p}(\mathbb{R}^N)$ and in $W^{1,p}(\Omega)$
I know that $C^\infty(\mathbb{R}^N) \cap W^{1,p}(\mathbb{R}^N)$ and $C^\infty_c (\mathbb{R}^N)$ are dense in $W^{1,p}(\mathbb{R}^N)$ for all $1\le p < \infty$.I also know that $C^\infty(\Omega) \cap...
View ArticleExamine the convergence of the improper integral $\int_1^{\infty} f(x) dx$
Examine the convergence of the improper integral $\int_1^{\infty} f(x) dx$ where$f(x) = \frac{1}{x^3}$ if $x$ be rational $\geq 1$ and$f(x) = \frac{-1}{x^3}$ if $x$ be irrational $>1$.The solution...
View ArticleGeometric interpretation of implicit differentiation
It is well known that, given a function $f:\mathbb{R} \to \mathbb{R} $, $f'(x_0)$ can be interpreted as the slope of the tangent line to $f$ in $x_0$. What about curves of the form $F(x, y, c)=0$,...
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