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Expectation for Trinomial distribution
I am trying to understand the proof of $\mathrm{E}[xy]=n(n-1)p_1p_2$ where x,y have a trinomial distribution with pmf: $p(x,y) = \frac{n!}{x!y!(nāxāy)!}p_...
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Why do the interesting antihomomorphisms tend to be involutions?
Given a semigroup $S$, define that an antihomomorphism on $S$ is a function $$* :S \rightarrow S$$ satisfying $(xy)^* = y^*x^*.$ Examples abound. Consider...
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proof for construction of 60 degree angle
I have a construction from the game Euclidea, puzzle 4.2: The puzzle is given point $A$ and line $\overleftrightarrow{BC}$ (just the line -- neither point...
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How to plot a phase portrait for this system of differential equations?
I beg your help.. I'd like the phase portrait for this system: \begin{aligned} \frac{dx}{dt} &= x (7-x-2y) \\ \frac{dy}{dt} &= y (5-y-x) \end...
Read Journal![A long nasty limit problem: $\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$](https://ly.shilpakalanarail.gov.bd/uploads/a-long-nasty-limit-problem-lim-x-to-infty-left-8e-sqrt-large-x-x-x-1-x-1-8x-2-4x-ln-x-ln-2-x-4x-2-ln-x-ln-2-pi-right_720.jpg)
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A long nasty limit problem: $\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln 2\pi\right]$
Does the following limit admit a closed-form? $$\lim_{x \to \infty}\left[8e\,\sqrt[\Large x]{x^{x+1}(x-1)!}- 8x^2-4x \ln x - \ln^2 x - (4x + 2 \ln x) \ln ...
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show lambda is an eigenvalue of matrix A and find one eigenvector x
Hello Lovely people of the Overflow :) I am working on a homework assigntment for my linear algebra class and i am stumped on this pesky question which is...
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User Jiashen Tang - Mathematics Stack Exchange
Q&A for people studying math at any level and professionals in related fields
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Question on Euler's summation formula
Proof from Apostol: (Annotations mine) If $f$ has a continuous derivative $f'$ on the interval $[y, x]$, where $0 < y < x$, then $$\sum\limits_...
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How to find ratio of two Continuous random variables
A quiz is 1 hour long, and has two questions. The time you give to Q1 is represented by the random variable $X$ which has the following pdf: $$f_X(x)=12x(...
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Bounded Convergence Theorem Proof
If $(f_n)$ is a pointwise convergent, uniformly bounded sequence of measurable functions on and interval $I:[0,1]$, then: $$\begin{align*} \lim_{n\to\inft...
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