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Compute $ \min_{a,b,c \in \mathbb{R}} \int_{-1}^1 |e^x-a-bx-cx^2|^2dx $

The question is the following: Compute $$ \min_{a,b,c \in \mathbb{R}} \int_{-1}^1 |e^x-a-bx-cx^2|^2dx $$ I tried to expand the integral, and it gives me: $$ \int_{-1}^1 |e^x-a-bx-cx^2|^2dx= 2a^2+\frac{2}{3}(2ac+b^2)+\frac{2}{e}(a-2b+5c)-2e(a+c)+\frac{2}{5}c^2+\frac{1}{2}(e^2-e^{-2}) $$ Here I have no idea how to proceed. I also consider the following: we can rewrite $$ \min_{a,b,c \in \mathbb{R}} \int_{-1}^1 |e^x-f(x)|^2dx $$ where $$ f(x) = […]

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Disproving equality of cartesian products

We are to disprove the statement $X \times Y = Y \times X \iff X = Y$ but I can’t think of an example where this would be false. If $X = Y$, then wouldn’t the Cartesian product be the same in either direction?

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Find the number of solutions for $\underbrace{x*x*\ldots*x}_{x\text{ 10 times}}=\frac{1}{10}$ with a given binary operation.

On the interval $(-1, 1)$, consider the binary operation $$x*y=\dfrac{2xy+3(x+y)+2}{3xy+2(x+y)+3}$$ with $x, y \in (-1, 1)$. I have to find the number of solutions for the equation: $$\underbrace{x*x*\ldots*x}_{x\text{ 10 times}}=\frac{1}{10}$$ Finding the left-hand side would be incredibly painful, so I didn’t try that. I looked at the previous sub-point of this exercise and it looked […]

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Most powerful test for BETA(a,b)

A single observation $X$ is taken from a $BETA(a, b)$ distribution. Given the hypothesis $$ H_0 : a=b=1 \space vs\space H_1 : a=b=\frac{1}{2} $$ I want to find the Most Powerful test. Well, we know by the Neyman-Pearson lemma that: $$ \frac{f(\widetilde{x}/\theta_1)}{f(\widetilde{x}/\theta_0)} > k $$ to some $$ k > 0 $$ we will have […]

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Leibniz formula for $\pi$, is there any ways to relate the two proofs?

There are 2 common proofs of Leibniz formula $\frac{\pi}{4}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots$ Calculus proof: The first come by studying the power series $\sum_{n=0}^{\infty}(-x)^{n}(2n+1)^{-1}=\frac{x^{1}}{1}-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\ldots$ and then “take the limit $x\rightarrow 1^{-}$” (apply Abel’s theorem). The proof proceed by differentiating, then recognize that this is a geometric series, and then integrating again, it can be shown that this is just […]

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The value of parameter $a$ for which $\dfrac{ax^2+3x-4}{a+3x-4x^2}$ takes all real values for $x\in R$ are:

The value of parameter $a$ for which $\dfrac{ax^2+3x-4}{a+3x-4x^2}$ takes all real values for $x\in R$ are: My question is why we need to validate end points i.e. $1,7$ which makes Discriminant = 0? (Refer the last part of my attempt) My attempt is as follows:- $$y=\dfrac{ax^2+3x-4}{a+3x-4x^2}$$ $$ya+3yx-4yx^2=ax^2+3x-4$$ $$x^2(-4y-a)+x(3y-3)+ya+4=0$$ As $x$ can be any real, so […]

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Variable reduction in a Linear Programming problem

I am currently working on a problem where I need to find a feasible solution to linear equality constraints. $\hspace{50mm}Ax=b,$ $\hspace{50mm}x>=0,$ $\hspace{38mm}$where $A \in R^{m*n}, b\in R^m, A_{ij} \in \{0,1\}$ $ \hspace{50mm}m << n$ The number of variables is greater than the number of constraints in orders of magnitude. The simplex method gives Basic Feasible […]

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Metric space proof

I’m learning calculus and we have the following problem in our book: $(X,d)$ is a metric space. Let $ \emptyset \neq A_1 \subset A_2 \subset X$. Prove that if $A_2$ is a bounded set, then $A_1$ is also a bounded set and the diamater of $A_1 \leq $ diameter of $A_2$ We would usually prove […]

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Examples for objects which are either finite or uncountable? (like a $\sigma$-algebra)

We know that $\sigma$-algebras are either finite or uncountable (perhaps because their definition allows us to build uncountably many different new sets out of (disjoint) countably many, which we can choose out of an infinite $\sigma$-algebra). another example would be the power set $\mathcal{P}(S)$ for any set $S$. What are other interesting examples from other […]

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Inequalities giving lower bound on variance

I’m interested in some inequalities giving bounds on variance, $\sigma^2$. For example Popoviciu’s inequality says If $m,M$ are the minimum and maximum values of a random variable $X$ with some underlying distribution, then $$\sigma^2 \leq \frac{(M-m)^2}{4}.$$ Similarly, the von Szokefalvi Nagy inequality gives a lower bound when the sample size, $n$ is finite. In that […]