# All Questions

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### How do I find these bounds using Chebychev's inequality and the Central Limit Theorem?

Let $X$ have gamma distribution with parameters $\alpha=7$ and $\lambda=1$. Investigate the value of $F_X(10)$ using these methods: Find a lower bound using Chebychev's inequality. Approximate the ...
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### Ratio for home work

In the year 2000 a rich aunty won £12000 and decided to share her money between her 3 nieces in the ratio of their age 14 10 18. she died in 2006 how much will each niece get?
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### posterior density for bayesian estimations

Suppose that the number of accidents occurring daily in a certain plant has a Poisson distribution with an unknown mean $\lambda$. Based on previous experience in similar industrial plants, suppose ...
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### Hom between 2 schemes

Why is the set $Hom(X,Y)$ between 2 schemes $X$ and $Y$ a scheme as well? Where can I read the construction? For example, $Hom(\mathbb{A}^1,\mathbb{A}^1)$ is the set of all polynomial, and what is the ...
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### subgame perfect nash equilibrium for war of attrition

the question is as follow: suppose that two players are playing war of attrition, that means both of them could choose either to fight or quit, if either one of them quit, the game ends, and if ...
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### Power series representation of $x$?

This may not be a very good question, but I'm totally stumped. I need to know the power series representation of $x$, or if there even is one. I'll show you why: I am trying to solve $y''+2xy'-y=x$ ...
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### Prove min(L) = all words in L that they don't have any prefix of themselves in L

We define the minimal words language of $L, \min(L)$, to be the language of all words in $L$ that don't have any prefix in $L$. Assume $L$ is regular language. I need to prove by building an ...
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### arithmetic mean of a sequence converges

We had a theorem that the means of a sequence also converges: Let $(a_n)_{n\in\mathbb N}$ be a convergent sequence. Then $\displaystyle \overline a_n=\sum_{k=1}^n \frac{a_k}n$ also converges. ...
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### Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k$?

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k\;$? I'm aware that $3 = 3^1$ but I would expect $3\cdot 3^k\;$ to be $\;9^k$ or $\;9^{k+1}$.
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### How do Flatlanders represent Möbius strips?

There are 3D representations of Klein bottles that give people in our 3D universe a pretty good idea of how one is constructed: We can sort of see how this thing needs to be 'twisted' in the fourth ...
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### Removing one edge from Graph

How many new graphs that are not isomorphic will I have by removing any of its edges (but only one!) ? I did following: Where the numbers mean which graph will I get by removing corresponding ...
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### $(a_{2n})$ and $(a_{2n+1})$ converges then $(a_n)$ converges

Whe had the following theorem in class: If $(a_{2n})_{n\in\mathbb N}$ and $(a_{2n+1})_{n\in\mathbb N}$ are convergent sequences with the same limit $a$, then the sequence $(a_{n})_{n\in\mathbb N}$ ...
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### Finding generators for an ideal consisting of the set of functions vanishing on a subset of Spec

Let $U$ be the union of the $x,y,z$ axes in complex affine 3-space. The set of functions that vanish on $U$ is an ideal. Can we neatly express their generators? There are a lot of similar such ...
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### Use three 11's and various math symbols to make an equation equal to 6

The puzzle is to use the following symbols $$+,\;-,\;*,\;/,\;(\;,\;),\;!, \;\sqrt(\cdot)$$ in order to make a valid equation out of $$11~~~~~~11~~~~~~~11 = 6.$$ (There are three elevens with space in ...
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### How to show that: $\gcd \Big(2^{2^a}+1 , 2^{2^b}+1\Big)=1$

How to show that: $$\gcd \Big(2^{2^a}+1 , 2^{2^b}+1 \Big)=1$$
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### Functions of bounded variation as the dual of $C([a,b])$

I am trying to understand this proposition about the dual of $C([a,b])$. I would like some help with the following: (1) What does the integral with respect to a function of bounded variation mean? ...
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### Intersection of monomial ideals

Let $[n] = \lbrace 1,2,\dots,n \rbrace$, and $F \subset [n]$. We denote by $P_F \subset K[X_1,\dots,X_n]$ the monomial ideal generated by the variables $X_i$ with $i \in F$. Given an ...
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### Probability of ace of spades on 21st selection

A deck of cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the 20th card to appear, what is the conditional prob that the ...
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### How to show that $f\left( \frac{ x + y }{ 2 }\right ) \leq \frac{ f( x ) + f( y ) }{ 2 }$ when $f''(x) \geq 0$.

I need to show that if $f: (a,b) \to \mathbb{ R }\text{ with}\;\; f''( x ) \geq 0$ for all $x \in (a,b)$, then $f\left( \frac{ x + y }{ 2 } \right) \leq \frac{ f( x ) + f( y ) }{ 2 }$. I know that ...
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### Men on a boat problem

There is the usual question of some men on a boat- various men have various speeds, the boat has a capacity of 2 men, and the boat takes on the speed of the slowest man in the boat at any given time. ...
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### dimension of a quotient of a local ring

Let $\mathcal{O}_{k^2,(0,0)}$ be the local ring at origin, i.e. $k[x,y]_{(x,y)}$. I want to show that $\dim_k \mathcal{O}_{k^2,(0,0)}/(y-x^2,x^3)=3$. My rough argument is the following, but I feel ...
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### Does $X\overset{f}{\rightarrow} Y\rightarrow0$ being exact imply $\operatorname{coker}f=0$?

In a category with zero object, it is easy to see that if $0\rightarrow X\overset{f}{\rightarrow} Y$ is exact then $\ker f=0$, since $0\rightarrow X$ is monic and hence is its own image. However, when ...
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### Limit question: $\lim_{x\to \infty}f(x),\;\; \lim_{x\to -\infty}f(x)$?

Assuming $$f(x)=ax^3+bx^2+cx+d$$ show that $\lim_{x\rightarrow\infty}f(x)$ and $\lim_{x\rightarrow -\infty}f(x)$ exists, find the limits.$a,b,c,d\in\mathbb{R}$ Well, I think the limits exists in ...
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### Losing at Spider Solitaire

Spider Solitaire has the property that sometimes none of the cards in the final deal can "go" and so you lose, regardless of how much progress you have made beforehand. You would have known that you ...
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### Am I doing something wrong with these moment generating functions?

I'm told to find the moment generating function of the pdf $6x(1-x)$, $0<x<1$, and I found $$\frac{6te^t-12e^t+6t+12}{t^3}$$ Then it asks to find the expected value, and as anyone else would ...