Applied Mathematics

Let [math]X_i, i \geq 1[/math] be independent with [math] P( X _i = 1) =p, P(X_i =-1) =q = 1-p[/math] with [math]S_n= \displaystyle\sum_{i=1}^n X_i[/math] Show that [math](\frac{q}{p})^{S_n}, n \geq 1[/math] is a martingale with mean 1. Solution: Let's prove this step-by-step: 1) The [math]X_i[/math] are independent random variables with: [math]P(X_i = 1) = p[/math] [math]P(X_i = -1) = q = 1 - p[/math] 2) Let [math]S_n = \displaystyle\sum_{i=1}^n X_i[/math] be the partial sums. 3) Consider the stochastic process [math]Y_n = (\frac{q}{p})^{S_n}[/math]. 4) To show this is a martingale, we need to prove: [math]E[Y_{n+1} | Y_1, ..., Y_n] …

This is from memory of high school physics. It concerns the mathematical relationship directly proportional to between force (F), mass (m) and acceleration (a). F = ma 1. F [math]\propto[/math] m 2. F [math]\propto[/math] a Therefore, says my physics textbook, 3. F [math]\propto[/math] ma Can someone please tell me how that's derived i.e. how do we infer 3 from 1 and 2? Muchas gracias in advance!

I have been thinking about how you could probe within a map or function in algebra and decided to try to use a sort of "probe" set in-between the sets that are being mapped together. It reminds me of composition and am trying it from that angle, however I need some help with it. The reason I want to do this is because I want to see if I can "discover" a mapping without the need of a mapping(for higher order math's) or function equation for algebra right now. The name could use some work for the topic.

Let E be a finite nonempty set and let [math]\Omega := E^{\mathbb{N}}[/math] be the set of all E-valued sequences [math] \omega = (\omega_n)_{n\in \mathbb{N}}[/math]. For any [math] \omega_1, \dots,\omega_n \in E[/math] Let [math][\omega_1, \dots,\omega_n]= \{\omega^, \in \Omega : \omega^,_i = \omega_i \forall i =1,\dots,n \}[/math] be the set of all sequences whose first n values are [math]\omega_1,\dots, \omega_n[/math]. Let [math]\mathcal{A}_0 =\{\emptyset\}[/math] for [math]n\in \mathbb{N}[/math] define [math]\mathcal{A}_n :=\{[\omega_1,\dots,\omega_n] : \omega_1,\dots, \omega_n \in E\}[/math]. Hence show that [math]\mathcal{A}= \bigcup_…

Parliament contains a proportion p of Labour members, who are incapable of changing their minds about anything, and a proportion 1 − p of Conservative members who change their minds completely at random (with probability r) between successive votes on the same issue. A randomly chosen member is noticed to have voted twice in succession in the same way. What is the probability that this member will vote in the same way next time? My answer: Let's solve this problem step by step. First, let's find the probability that the member is a Labour member given that they have voted twice in succession in the same way. We can use Bayes' theorem for this. Let L…

My answers: 1. - n (n-1) [imath] -\displaystye\frac{n(n-1)(n-2)}{27}[/imath] 2.

Hello. I'm not a mathematician and I'm designing a sailboat hull. I have to test the shape of the underwater portion of the hull, at different hell angle ( angle of the hull in relation to the plane representing the water surface). To do so, I have the constraint to keep constant the volume of the underwater portion of the hull at different heel angles. So I'm using a polynomial curve fitting approach where I have the X value representing the depth of the submerge portion of the hull and the Y value corresponding to the volume of the submerge part . I sample 100 of these x,y values on my drawing and I fit curves of different degrees . To test which degree is the best…

Emma makes a straight prism with an equilateral triangle as its base from a net labeled as shown in the adjacent figure. She uses the prism as a 'thrower, where the letter at the bottom is considered to be rolled. Show: If the probability of rolling M A T H E is greater than the probability of rolling EM M A, then the probability of rolling T E E is greater than 1/64 Note: Rolling M A T H E means, if the die is rolled five times in this order, rolling the M. A, T H, and E to be rolled. Similarly, "E M M A roll" and "T E E roll" are for four and four times respectively.

Here, I present a few silly doubts on how to define the maximum number of solutions of a polynomial using set notation and theory. Let Qn(x) be the inverse of an nth-degree polynomial. Precisely, Qn(x)=|R(x)/Pn(x)|. Here, it is of my interest to define a number, J(n), that provides the maximum number of singular points of Qn(x) or the maximum number of solutions of Pn(x)=0 since (Qn(x))-1 equals Pn(x). To that end, I have tried to create the following definition: J(n)=Sup{\gamma((Qn(x))-1=0):dQ \leq n} in which \gamma((Qn(x))-1=0) denotes the number of solutions of Pn(x)=0 and d indicates the degree of Q. Based on the above, I ask: 1. Is the above definition corre…

If sports team A is expected to score between 10-20 points, and team b between 17-24 points, what is the calculated percentage that team B will win by 3 points or more. Basically, I'm looking to determine the total number of possibilities team B will win by 3 or more points, and what is the total number of possible scores. Also I need the formulas so I can change the three variables. Thank you, GD

Now my son wants a statistics textbook. (for the uninitiated, I am only his humble scribe and mother). I remember my stats textbooks from 35 years ago in grad school (Statistics: An Intuitive Approach: Weinberg, George H.: 9780818504266: Amazon.com: Books), but they are not necessarily the best today. He wants one that will take him from relatively basic stuff up to regression analysis. If not one, then two in a series. Hopefully something that is inexpensive and/or available secondhand through Amazon. Suggestions? Thank you.

There is an axis for Imaginary numbers, and there is an axis for real numbers. So we can just use the X and Y axes, what is the difference? They can do the same thing.

Binary is a base-2 number system that uses two states 0 and 1 to represent a number. Does it mean it only use 2 digits. So decimal is a bse-10 number system.

have a look at the short paper attached. The number of the equal triangles should be 1+n/(n/2), if n=10, it is 55. Of you also count those up side down equal triangles, it should be n^2, if n = 10, it is 100. Then what is the author talking about?

binary is "base-2" system. What does "base-2" mean? Is it -- all numbers are in the form 2^x?

First time posting so apologies if this question is in the wrong section. I've recently gotten into the cryptocurrency market and so I've spent a lot of time online studying market analysis. When i got started i though it was the study of how stable a business or commodity is, outlook on productions, etc... instead I find that "Market Analysis" is the study of previous market movements to determine future market movements. some examples can be found here: https://optionalpha.com/13-stock-chart-patterns-that-you-cant-afford-to-forget-10585.html I've studied this hard and deep and then came to a sudden realization - This seems an awful lot like horoscopes.…

When you look at the world around you, you'll be amazed. It was the Chinese gene that took over the world fastest, and spawned fastest. (largest population in the world: China) Is that because A) the first humans had more Chinese genome in their DNA than, say, the Mexican gene, leading to evolution patterns that created a greater Chinese population than Mexican population, or B) because the Chinese gene is more evolutionary-savvy than the Mexican gene? but If A), then why did the first humans not have genes of different races in equal proportions? or If B), that it is true that some genes are more evolutionary-savvy than others, t…

https://vixra.org/abs/2201.0186 Hi every body , here a solotion for the Collatz conjecture. If there is any comment , question or critic i ll be happy to discuss it (even if i am more frocophone than anglophone)

Hi everyone, I hope someone will be kind enough to approve my article titled: Probabilistic solution for the Syracuse conjecture on Arxiv (combinatorics and number theory). Here is the link for the article. Thank you very much in advance Sorry the link!!!!! (deleted)

The Copernican Principle, officially described by Richard Gott (J. Richard Gott III | Department of Astrophysical Sciences (princeton.edu)) in 1993, basically says, "I'm not special." It can be applied for statistical predictions in cases where our place in space or time may be considered random. E.g. the fact that I was born in this specific time in the history of human civilization seems to be random, i.e. I could be equally likely born in any other time of that history. Now, the human civilization, with towns, writing, some kind of machines, etc. exists about 10,000 years. My random place in it is - with 95% confidence - inside the middle 95% interval of its life…

0/0 = 0, means 0 X 0 = 0. It is well defined.

Please give me a simple proof, X^0=1. Thanks.

See example 3 at https://math.berkeley.edu/~ehallman/summer-2015/problemSheet6-2-sols.pdf Why the answer is 105/221 and not (3/8)(40/100)/(3/8(40/100)+2/9(30/100)+(6/7)(30/100))=63/199?

I want to develop a 2D random field and its change with time with constant velocity. My process: 1. Define a 2D grid (not the fields yet) [x,y] with n×n points 2. Define 1D time axis [t] with nt elements 3. Find the Lagrangian distance D between the points in space and with time [ with advection velocities in x and y , i.e., [vx,vy] ]. For now vx and vy are constants. D[t,(xi,yi),(xj,yj)]=(xi−vxt−xj)2+(yi−vyt−yj)2−−−−−−−−−−−−−−−−−−−−−−−−−−√ D has size [nt×n2×n2] 4. Use this distance matrix and the time axi…