• Math

    A Thing About the Hot Hand Fallacy and the “Law of Small Numbers”

    There was an interesting post and discussion on the NBA subreddit of Reddit on the Hot Hand phenomenon and whether or not it is a fallacy. Comment from discussion RyanLeafgOaT8’s comment from discussion "[OC] Going Nuclear: Klay Thompson’s Three-Point Percentage after Consecutive Makes". A Numberphile video on the topic: An article on the topic: https://www.scientificamerican.com/article/do-the-golden-state-warriors-have-hot-hands/ In some parts of the Numberphile video, Professor Lisa Goldberg emphasizes that issues of the “Law of Small Numbers,” which is described in the Scientific American article as: Early in their careers, Amos Tversky and Daniel Kahneman considered the human tendency to draw conclusions based on a few observations, which they called the ‘‘law of…

  • Economics,  Math

    The Terms of Trade of Brazil

    Source: https://www.nytimes.com/2018/11/09/opinion/what-the-hell-happened-to-brazil-wonkish.html An article in the New York Times by Paul Krugman talked about a current economic downturn in Brazil. What happened: First, the global environment deteriorated sharply, with plunging prices for the commodity exports still crucial to the Brazilian economy. Second, domestic private spending also plunged, maybe because of an excessive buildup of debt. Third, policy, instead of fighting the slump, exacerbated it, with fiscal austerity and monetary tightening even as the economy was headed down. What didn’t happen: Maybe the first thing to say about Brazil’s crisis is what it wasn’t. Over the past few decades those who follow international macroeconomics have grown more or less accustomed to…

  • Math

    Test Coin2

    https://math.stackexchange.com/questions/2033370/how-to-determine-the-number-of-coin-tosses-to-identify-one-biased-coin-from-anot/2033739#2033739   Suppose there are two coins and the percentage that each coin flips a Head is \(p\) and \(q\), respectively. \(p, q \in [0,1] \), \(p \neq q \), and the values are given and known. If you are free to flip one of the coins any number of times, how many times \(n\) do you have to flip the coin to decide with some significance level \( \left( \textrm{say } \alpha = 0.05 \right) \) that it’s the \(p\) coin or the \(q\) coin that you’ve been flipping? The distribution of heads after \(n\) flips for a coin will be a binomial distribution with means at \(pn\) and…

  • Math

    Test Coin

    https://math.stackexchange.com/questions/2033370/how-to-determine-the-number-of-coin-tosses-to-identify-one-biased-coin-from-anot/2033739#2033739 Suppose there are two coins and the percentage that each coin flips a Head is \(p\) and \(q\), respectively. \(p, q \in [0,1] \) and the values are given and known. If you are free to flip one of the coins, how many times \(n\) do you have to flip the coin to decide with some significance level \( \left( \textrm{say } \alpha = 0.05 \right) \) that it’s the \(p\) coin or the \(q\) coin that you’ve been flipping? The distribution of heads after \(n\) flips for a coin will be a binomial distribution with means at \(pn\) and \(qn\). The Usual Hypothesis Test In the usual hypothesis…

  • Math

    Barcodes and Modular Arithmetic

    Barcodes Here is an example of a UPC-A barcode, taken from wikipedia: A UPC-A barcode has 12 digits.  The first digit is something that tells how the numbers are generally used – for example, a particular industry might use a certain number for certain kinds of items.  The last twelfth digit is a check digit that can try to tell whether or not the numbers have an error.  This check digit is constructed in a certain way at first.  Later on, the check digit may be able to tell us if the numbers have an error or not.   The check digit is constructed as follows: We have 11 digits:…

  • Economics,  Math

    Portfolio Insurance and Black Monday, October 19, 1987

    On the thirtieth anniversary of Black Monday, the stock market crash of October 19th and 20th in 1987, there have been mentions of “portfolio insurance” having possibly exacerbated the crash.   Portfolio insurance, in principle, is exactly what you might expect it to be: if you own a stock, Stock A, you insure it with a put option on Stock A.  Your position becomes equivalent to a call option on Stock A until the put option expires, with the price of this position being the premium of the put option when you bought it. If you are managing a portfolio on behalf of clients, though, and you just need to…

  • Math

    Testing MathJax-LaTeX

    emph https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference Default font size. Different font size. Bold text. bold text. italic text. Italic text. Underlined text. At first, we sample $$f(x)$$ in the \(N\) ($N$ is odd) equidistant points around \(x^*\): \[ f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \] where \(h\) is some step. Then we interpolate points \((x_k,f_k)\) by polynomial \begin{equation} \label{eq:poly} \tag{1} P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j} \end{equation} Its coefficients \(a_j\) are found as a solution of system of linear equations: \begin{equation} \label{eq:sys} \tag{asdf} \left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \end{equation} \begin{equation} \label{eq:sys2} \tag{asdf2} \{ P_{N-1}(x_k) = f_k\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \end{equation} Backslash left and right parentheses: \[ \left( \frac{1}{2} \right) \qquad ( \frac{1}{2} ) \\ ( \frac{1}{2} ) \] $$ 1…

  • Math

    Brainteaser: The Monty Hall Problem

    You are on a game show and presented with 3 doors.  Behind one is a car and behind the other 2 are goats.  You want to choose the door with a car behind it, as if you do so, you win the car.  You choose one door.  Then, the host opens one of the other doors, which reveals a goat behind it.  The host gives you a choice to either switch your door to the other one that’s still closed or keep your original choice.  Should you switch doors?     Answer: If your strategy is to stick to your original choice, your probability of choosing the door with the…

  • Math

    Brainteaser: 100 Prisoners in a Line and Extension

    There are 100 prisoners.  An executioner tells them that tomorrow morning, he will line them up so that each of them is facing the back of the head of another prisoner, except for one prisoner at the end of the line.  In other words, prisoner 1 sees the back of the head of prisoner 2 as well as the backs of the heads of prisoners 3-100, prisoner 2 sees the back of the heads of prisoners 3-100, …, prisoner 99 only sees the back of the head of prisoner 100, and prisoner 100 doesn’t see any prisoners in front of him.  The executioner tells them that he will put either…

  • Math

    Brainteaser: Blue Foreheads

    100 people are in a room.   All 100 of them are perfect logicians. They are told that at least one person in the room has blue paint on their forehead. They are told that once you deduce that you have blue paint on your forehead, the next time that the lights are turned off, leave the room.   All 100 people have actually had their foreheads painted blue (but of course, each of them don’t know this at this point – they can only see the other people’s foreheads).  The light is turned off, then on, then off, on, etc.  What happens?     Answer: So each person sees…