What is the probability that no more than four browsing customers will buy something during a specified hour?

Shahida Akter
2022-07-26

What is the probability that no more than four browsing customers will buy something during a specified hour?

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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 2022-08-10

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 2022-07-21

If I found that a series converges, how can I know to what number it's converging to?

I started learning series in calculus and I have trouble catching a basic concept. When I try to find if a series converges or diverges I have many ways to go about it. If I see that the series diverges than I stop there. If I see that the series converges than there is a number it's converging to right?

For example: $\sum \frac{2}{{n}^{3}+4}$. I do the limit comparison test with the series $\sum \frac{1}{{n}^{3}}$ and get a finite number 2. I know that $\sum \frac{1}{{n}^{3}}$ converges, so now I know that $\sum \frac{2}{{n}^{3}+4}$ converges also. How do I know to what number it converges to?

I started learning series in calculus and I have trouble catching a basic concept. When I try to find if a series converges or diverges I have many ways to go about it. If I see that the series diverges than I stop there. If I see that the series converges than there is a number it's converging to right?

For example: $\sum \frac{2}{{n}^{3}+4}$. I do the limit comparison test with the series $\sum \frac{1}{{n}^{3}}$ and get a finite number 2. I know that $\sum \frac{1}{{n}^{3}}$ converges, so now I know that $\sum \frac{2}{{n}^{3}+4}$ converges also. How do I know to what number it converges to?

asked 2022-08-09

Study of an infinite product

During some research, I obtained the following convergent product ${P}_{a}(x):=\prod _{j=1}^{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\left(\frac{x}{{j}^{a}}\right)\phantom{\rule{1em}{0ex}}(x\in \mathbb{R},a>1).$.

Considering how I got it, I know it's convergent and continuous at 0 (for any fixed a), but if I look at ${P}_{a}$ now, it doesn't seem so obvious for me.

Try: I showed that ${P}_{a}\in {L}^{1}(\mathbb{R})$, i.e. it is absolutely integrable on $\mathbb{R}$. Indeed, by using the linearization of the cosine function and the inequalities $\mathrm{ln}(1-y)\u2a7d-y$ (for any $y<1$) and $1-\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}z\u2a7e{z}^{2}/2$ (for any real z), we obtain

$\begin{array}{rcl}{P}_{a}(x{)}^{2}& =& \prod _{j=1}^{\mathrm{\infty}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\prod _{j>|x{|}^{1/a}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\mathrm{exp}(-C|x{|}^{1/a}),\end{array}$ for some absolute constant $C>0$. However, I have not been able to use this upper bound to prove continuity at 0 (via uniform convergence for example, if we can).

Question : I wanted to know how to study ${P}_{a}$ (e.g. its convergence and continuity at 0), if you think it has an other form "without product" (or other nice properties) and finally if anyone has already seen this type of product (in some references/articles), please.

During some research, I obtained the following convergent product ${P}_{a}(x):=\prod _{j=1}^{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\left(\frac{x}{{j}^{a}}\right)\phantom{\rule{1em}{0ex}}(x\in \mathbb{R},a>1).$.

Considering how I got it, I know it's convergent and continuous at 0 (for any fixed a), but if I look at ${P}_{a}$ now, it doesn't seem so obvious for me.

Try: I showed that ${P}_{a}\in {L}^{1}(\mathbb{R})$, i.e. it is absolutely integrable on $\mathbb{R}$. Indeed, by using the linearization of the cosine function and the inequalities $\mathrm{ln}(1-y)\u2a7d-y$ (for any $y<1$) and $1-\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}z\u2a7e{z}^{2}/2$ (for any real z), we obtain

$\begin{array}{rcl}{P}_{a}(x{)}^{2}& =& \prod _{j=1}^{\mathrm{\infty}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\prod _{j>|x{|}^{1/a}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\mathrm{exp}(-C|x{|}^{1/a}),\end{array}$ for some absolute constant $C>0$. However, I have not been able to use this upper bound to prove continuity at 0 (via uniform convergence for example, if we can).

Question : I wanted to know how to study ${P}_{a}$ (e.g. its convergence and continuity at 0), if you think it has an other form "without product" (or other nice properties) and finally if anyone has already seen this type of product (in some references/articles), please.

asked 2022-07-14

Probability of 1 billion monkeys typing a sentence if they type for 10 billion years

Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These monkeys type for 10 billion years. What is the probability that they can type the first sentence of Lincoln’s “Gettysburg Address”?

Four score and seven years ago our fathers brought forth on this continent a new nation conceived in liberty and dedicated to the proposition that all men are created equal.

Hint: Look up Boole’s inequality to provide an upper bound for the probability!

This is a homework question. I just want some pointers how to move forward from what I have done so far. Below I will explain my research so far.

First I calculated the probability of the monkey 1 typing the sentence (this question helped me do that); let's say that probability is p:

$P(\text{Monkey 1 types our sentence})=P({M}_{1})=p$

Now let's say that the monkeys are labeled ${M}_{1}$ to ${M}_{{10}^{9}}$, so given the hint in the question I calculated the upper bound for the probabilities of union of all $P({M}_{i})$ (the probability that i-th monkey types the sentence) using Boole's inequality.

Since $P({M}_{i})=P({M}_{1})=p$,

$P\left(\bigcup _{i}{M}_{i}\right)\le \sum _{i=1}^{{10}^{9}}P({M}_{i})=\sum ^{{10}^{9}}p={10}^{9}\phantom{\rule{thinmathspace}{0ex}}p$

Am I correct till this point? If yes, what can I do more in this question? I tried to study Bonferroni inequality for lower bounds but was unsuccessful to obtain a logical step. If not, how to approach the problem?

Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These monkeys type for 10 billion years. What is the probability that they can type the first sentence of Lincoln’s “Gettysburg Address”?

Four score and seven years ago our fathers brought forth on this continent a new nation conceived in liberty and dedicated to the proposition that all men are created equal.

Hint: Look up Boole’s inequality to provide an upper bound for the probability!

This is a homework question. I just want some pointers how to move forward from what I have done so far. Below I will explain my research so far.

First I calculated the probability of the monkey 1 typing the sentence (this question helped me do that); let's say that probability is p:

$P(\text{Monkey 1 types our sentence})=P({M}_{1})=p$

Now let's say that the monkeys are labeled ${M}_{1}$ to ${M}_{{10}^{9}}$, so given the hint in the question I calculated the upper bound for the probabilities of union of all $P({M}_{i})$ (the probability that i-th monkey types the sentence) using Boole's inequality.

Since $P({M}_{i})=P({M}_{1})=p$,

$P\left(\bigcup _{i}{M}_{i}\right)\le \sum _{i=1}^{{10}^{9}}P({M}_{i})=\sum ^{{10}^{9}}p={10}^{9}\phantom{\rule{thinmathspace}{0ex}}p$

Am I correct till this point? If yes, what can I do more in this question? I tried to study Bonferroni inequality for lower bounds but was unsuccessful to obtain a logical step. If not, how to approach the problem?

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 2022-08-11

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 $\cdot :G\times G\times G\to G$