From $10$ numbers $a,b,c,...j$ all sets of $4$ numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the $10$ numbers?

posader86
2022-07-23
Answered

From $10$ numbers $a,b,c,...j$ all sets of $4$ numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the $10$ numbers?

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encoplemt5

Answered 2022-07-24
Author has **15** answers

There are $C(10,4)=\frac{10!}{4!6!}$ different sets of $4$ numbers chosen among ${x}_{1},{x}_{2},\dots {x}_{10}$. Each number ${x}_{i}$ belongs to $C(9,3)=\frac{9!}{3!6!}$ such sets, because the other three numbers in the same set can be chosen in $C(9,3)$ different ways. Hence the average on all sets is:

$\begin{array}{rl}\frac{1}{C(10,4)}\sum _{1\le i<j<k<l\le 10}\frac{{x}_{i}+{x}_{j}+{x}_{k}+{x}_{l}}{4}& =\frac{1}{4C(10,4)}\sum _{i=1}^{10}C(9,3){x}_{i}\\ & =\frac{C(9,3)}{4C(10,4)}\sum _{i=1}^{10}{x}_{i}=\frac{1}{10}\sum _{i=1}^{10}{x}_{i}\end{array}$

and both averages are the same.

This also works in general for the case of all sets of $n$ numbers chosen among $N$. The key is all numbers ${x}_{i}$ appear the same number of times in the final sum.

$\begin{array}{rl}\frac{1}{C(10,4)}\sum _{1\le i<j<k<l\le 10}\frac{{x}_{i}+{x}_{j}+{x}_{k}+{x}_{l}}{4}& =\frac{1}{4C(10,4)}\sum _{i=1}^{10}C(9,3){x}_{i}\\ & =\frac{C(9,3)}{4C(10,4)}\sum _{i=1}^{10}{x}_{i}=\frac{1}{10}\sum _{i=1}^{10}{x}_{i}\end{array}$

and both averages are the same.

This also works in general for the case of all sets of $n$ numbers chosen among $N$. The key is all numbers ${x}_{i}$ appear the same number of times in the final sum.

asked 2022-08-14

Linear multivariate recurrences with constant coefficients

In the theory of univariate linear recurrences with constant coefficients, there is a general method of solving initial value problems based on characteristic polynomials. I would like to ask, if any similar method is known for multivariate linear recurrences with constant coefficients. E.g., if there is a general method for solving recurrences like this:

$f(n+1,m+1)=2f(n+1,m)+3f(n,m)-f(n-1,m),\phantom{\rule{2em}{0ex}}f(n,0)=1,f(0,m)=m+2.$

Moreover, is their any method for solving recurrences in several variables, when the recurrence goes only by one of the variables? E.g., recurrences like this:

$f(n+1,m)=f(n,2m)+f(n-1,0),\phantom{\rule{2em}{0ex}}f(0,m)=m.$

This second question is equivalent to the question, if there is a method of solving infinite systems of linear univariate recurrences with constant coefficients. That is, using these optics, the second recurrence becomes ${f}_{m}(n+1)={f}_{2m}(n)+{f}_{0}(n-1),\phantom{\rule{2em}{0ex}}{f}_{m}(0)=m,\phantom{\rule{2em}{0ex}}m=0,1,\dots .$

I am not interested in a solution of any specific recurrence, but in solving such recurrences in general, or at least in finding out some of the properties of possible solutions. For instance, for univariate linear recurrences, each solution has a form ${c}_{1}{p}_{1}(n){z}_{1}^{n}+\dots +{c}_{k}{p}_{k}(n){z}_{k}^{n},$, where ${c}_{i}$'s are constants, ${p}_{i}$'s are polynomials and ${z}_{i}$'s are complex numbers. Does any similar property hold for some class of recurrences similar to what I have written?

I have been googling a lot, but have found only methods for some very special cases (in monographs on partial difference equations, etc.), but nothing general enough. I am not asking for a detailed explanation of any method, but references to the literature would be helpful. I don't know much about transforms (like discrete Fourier transform or z-transform), but I found certain hints that there could be a method based on these techniques. Is it possible to develop something general enough using transform, i.e., is the study of transforms worth an effort (in the context of solving these types of recurrences)? However, it seems to me that the generalization of the characteristic polynomial method (perhaps, some operator-theoretic approach) could lead to more general results. Has there been any research on this topic?

In the theory of univariate linear recurrences with constant coefficients, there is a general method of solving initial value problems based on characteristic polynomials. I would like to ask, if any similar method is known for multivariate linear recurrences with constant coefficients. E.g., if there is a general method for solving recurrences like this:

$f(n+1,m+1)=2f(n+1,m)+3f(n,m)-f(n-1,m),\phantom{\rule{2em}{0ex}}f(n,0)=1,f(0,m)=m+2.$

Moreover, is their any method for solving recurrences in several variables, when the recurrence goes only by one of the variables? E.g., recurrences like this:

$f(n+1,m)=f(n,2m)+f(n-1,0),\phantom{\rule{2em}{0ex}}f(0,m)=m.$

This second question is equivalent to the question, if there is a method of solving infinite systems of linear univariate recurrences with constant coefficients. That is, using these optics, the second recurrence becomes ${f}_{m}(n+1)={f}_{2m}(n)+{f}_{0}(n-1),\phantom{\rule{2em}{0ex}}{f}_{m}(0)=m,\phantom{\rule{2em}{0ex}}m=0,1,\dots .$

I am not interested in a solution of any specific recurrence, but in solving such recurrences in general, or at least in finding out some of the properties of possible solutions. For instance, for univariate linear recurrences, each solution has a form ${c}_{1}{p}_{1}(n){z}_{1}^{n}+\dots +{c}_{k}{p}_{k}(n){z}_{k}^{n},$, where ${c}_{i}$'s are constants, ${p}_{i}$'s are polynomials and ${z}_{i}$'s are complex numbers. Does any similar property hold for some class of recurrences similar to what I have written?

I have been googling a lot, but have found only methods for some very special cases (in monographs on partial difference equations, etc.), but nothing general enough. I am not asking for a detailed explanation of any method, but references to the literature would be helpful. I don't know much about transforms (like discrete Fourier transform or z-transform), but I found certain hints that there could be a method based on these techniques. Is it possible to develop something general enough using transform, i.e., is the study of transforms worth an effort (in the context of solving these types of recurrences)? However, it seems to me that the generalization of the characteristic polynomial method (perhaps, some operator-theoretic approach) could lead to more general results. Has there been any research on this topic?

asked 2022-08-10

How do I determine sample size for a test?

Say you have a die with n number of sides. Assume the die is weighted properly and each side has an equal chance of coming up. How do I determine the minimum number of rolls needed so that results show an equal distribution, within an expected margin of error?

I assume there is a formula for this, but I am not a math person, so I don't know what to look for. I have been searching online, but haven't found the right thing.

Say you have a die with n number of sides. Assume the die is weighted properly and each side has an equal chance of coming up. How do I determine the minimum number of rolls needed so that results show an equal distribution, within an expected margin of error?

I assume there is a formula for this, but I am not a math person, so I don't know what to look for. I have been searching online, but haven't found the right thing.

asked 2022-08-14

In an Math Exam there are $80$ more men than women. The result showed that the women's average is $20\mathrm{\%}$ higher than men's, and that the total average is $75\mathrm{\%}$. what is the women's average?

asked 2022-07-21

There is a box with $12$ dice which all look the same. However there are actually three types of dice:

$6$ normal dice. The probability to get a $6$ is $1/6$ for each dice.

$3$ biased dice. The probability to get a $6$ is $0.85$.$3$ biased dice. The probability to get a $6$ is $0.05$.You take a die from the box at random and roll it.What is the conditional probability that it is of type $b$, given that it gives a $6$?

$6$ normal dice. The probability to get a $6$ is $1/6$ for each dice.

$3$ biased dice. The probability to get a $6$ is $0.85$.$3$ biased dice. The probability to get a $6$ is $0.05$.You take a die from the box at random and roll it.What is the conditional probability that it is of type $b$, given that it gives a $6$?

asked 2022-08-12

In survey question, what is Nonresponse bias?

asked 2022-07-14

Calculating conditional probability given Poisson variable

I encountered a set of problems while studying statistics for research which I have combined to get a broader question. I want to know if this is a solvable problem with enough information specifically under what assumptions or approach.

Given that a minesweeper has encountered exactly 5 landmines in a particular 10 mile stretch, what is the probability that he will encounter exactly 6 landmines on the next 10 mile stretch. (Average number of landmines is 0.6 per mile in the 50 mile stretch)

I have figured that the approach involves finding out the Poisson probabilities of the discrete random variable with the combination of Bayes Conditional probability. But am stuck with proceeding on applying the Bayes rule. i.e $Pr(X=6\mid X=5)$.

I know that $Pr(X=5)={e}^{-6}{5}^{6}/5!.$ Here $\lambda =0.6\cdot 10$ and $X=5$) Similarly for $Pr(X=6)$. Is Bayes rule useful here: $P(Y\mid A)=Pr(A\mid Y)Pr(Y)/(Pr(A\mid Y)Pr(Y)+Pr(A\mid N)Pr(N))$?

Would appreciate any hints on proceeding with these types of formulations for broadening my understanding.

I encountered a set of problems while studying statistics for research which I have combined to get a broader question. I want to know if this is a solvable problem with enough information specifically under what assumptions or approach.

Given that a minesweeper has encountered exactly 5 landmines in a particular 10 mile stretch, what is the probability that he will encounter exactly 6 landmines on the next 10 mile stretch. (Average number of landmines is 0.6 per mile in the 50 mile stretch)

I have figured that the approach involves finding out the Poisson probabilities of the discrete random variable with the combination of Bayes Conditional probability. But am stuck with proceeding on applying the Bayes rule. i.e $Pr(X=6\mid X=5)$.

I know that $Pr(X=5)={e}^{-6}{5}^{6}/5!.$ Here $\lambda =0.6\cdot 10$ and $X=5$) Similarly for $Pr(X=6)$. Is Bayes rule useful here: $P(Y\mid A)=Pr(A\mid Y)Pr(Y)/(Pr(A\mid Y)Pr(Y)+Pr(A\mid N)Pr(N))$?

Would appreciate any hints on proceeding with these types of formulations for broadening my understanding.

asked 2022-08-11

Is anybody researching "ternary" groups?

As someone who has an undergraduate education in mathematics, but didn't take it any further, I have often wondered something.

Of course mathematicians like to generalize ideas. i.e. it is often better to define and write proofs for a wider scope of objects than for a specific type of object. A kind of "paradox" if you will - the more general your ideas, often the deeper the proofs (quoting a professor).

Anyway I used to often wonder about group theory especially the idea of it being a set with a list of axioms and a binary function $a\cdot b=c$. But has anybody done research on tertiary (or is that trinary/ternary) groups? As in, the same definition of a group, but with a $\cdot (a,b,c)=d$ function.

Is there such a discipline? Perhaps it reduces to standard group theory or triviality and is provably of no interest. But since many results in Finite Groups are very difficult, notably the classification of simple groups, has anybody studied a way to generalize a group in such a way that a classification theorem becomes simpler? As a trite example: Algebra was pretty tricky before the study of imaginary numbers. Or to be even more trite: the Riemann $\zeta $ function wasn't doing much before it was extended to the whole complex plane.

EDIT: Just to expand what I mean. In standard groups there is an operation $\cdot :G\times G\to G$ I am asking about the case with an operation $\cdot :G\times G\times G\to G$

As someone who has an undergraduate education in mathematics, but didn't take it any further, I have often wondered something.

Of course mathematicians like to generalize ideas. i.e. it is often better to define and write proofs for a wider scope of objects than for a specific type of object. A kind of "paradox" if you will - the more general your ideas, often the deeper the proofs (quoting a professor).

Anyway I used to often wonder about group theory especially the idea of it being a set with a list of axioms and a binary function $a\cdot b=c$. But has anybody done research on tertiary (or is that trinary/ternary) groups? As in, the same definition of a group, but with a $\cdot (a,b,c)=d$ function.

Is there such a discipline? Perhaps it reduces to standard group theory or triviality and is provably of no interest. But since many results in Finite Groups are very difficult, notably the classification of simple groups, has anybody studied a way to generalize a group in such a way that a classification theorem becomes simpler? As a trite example: Algebra was pretty tricky before the study of imaginary numbers. Or to be even more trite: the Riemann $\zeta $ function wasn't doing much before it was extended to the whole complex plane.

EDIT: Just to expand what I mean. In standard groups there is an operation $\cdot :G\times G\to G$ I am asking about the case with an operation $\cdot :G\times G\times G\to G$