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Trigonometry

June 22, 2012 Leave a comment

I have just realized that I haven’t posted a good puzzle here in a long time, so here it goes one on Trigonometry, that the average student of Calculus should be able to tackle: you can use anything you think it could help: derivatives, symmetry, periodicity, integration, summation, go to several variables, differential equations, etc

Prove that, for all real values x \in \mathbb{R}, it is

\sin x = x \cos\big(\tfrac{x}{2}\big) \cos\big(\tfrac{x}{4}\big) \cos\big(\tfrac{x}{6}\big) \cos\big(\tfrac{x}{8}\big) \dotsb \cos\big( \tfrac{x}{2n}\big) \dotsb

or, in a more compact notation,

\sin x = x \displaystyle{\prod_{n=1}^\infty \cos \big( \tfrac{x}{2n} \big)}

Naïve Bayes

June 21, 2012 Leave a comment

There is nothing naïve about Naïve Bayes—a very basic, but extremely efficient data mining method to take decisions when a vast amount of data is available. The name comes from the fact that this is the simplest application to this problem, upon (the naïve) assumption of independence of the events. It is based on Bayes’ rule of conditional probability: If you have a hypothesis H and evidence E that bears on that hypothesis, then

\mathrm{Pr} \big( H \lvert E \big) = \displaystyle{ \frac{\mathrm{Pr} \big( E \lvert H\big) \mathrm{Pr}(H)}{\mathrm{Pr}(E)} }

where as usual, \mathrm{Pr}(A) denotes the probability of the event A, and \mathrm{Pr}\big( A \lvert B \big) denotes the probability of the event A conditional to another event B.

I would like to show an example of this technique, of course, with yet another decision-making algorithm oriented to guess my reaction to a movie I have not seen before. From the data obtained in a previous post, I create a simpler table with only those movies that have been scored more than 28 times (by a pool of 87 of the most popular critics featured in www.metacritics.com) [I posted the script to create that table at the end of the post]

Let’s test it:

>>> table=prepTable(scoredMovies,28)
>>> len(table)

49
>>> [entry[0] for entry in table]

[‘rabbit-hole’, ‘carnage-2011’, ‘star-wars-episode-iii—revenge-of-the-sith’,
‘shame’, ‘brokeback-mountain’, ‘drive’, ‘sideways’, ‘salt’,
‘million-dollar-baby’, ‘a-separation’, ‘dark-shadows’,
‘the-lord-of-the-rings-the-return-of-the-king’, ‘true-grit’, ‘inception’,
‘hereafter’, ‘master-and-commander-the-far-side-of-the-world’, ‘batman-begins’,
‘harry-potter-and-the-deathly-hallows-part-2’, ‘the-artist’, ‘the-fighter’,
‘larry-crowne’, ‘the-hunger-games’, ‘the-descendants’, ‘midnight-in-paris’,
‘moneyball’, ‘8-mile’, ‘the-departed’, ‘war-horse’,
‘the-lord-of-the-rings-the-fellowship-of-the-ring’, ‘j-edgar’,
‘the-kings-speech’, ‘super-8’, ‘robin-hood’, ‘american-splendor’, ‘hugo’,
‘eternal-sunshine-of-the-spotless-mind’, ‘the-lovely-bones’, ‘the-tree-of-life’,
‘the-pianist’, ‘the-ides-of-march’, ‘the-quiet-american’, ‘alexander’,
‘lost-in-translation’, ‘seabiscuit’, ‘catch-me-if-you-can’, ‘the-avengers-2012’,
‘the-social-network’, ‘closer’, ‘the-girl-with-the-dragon-tattoo-2011’]
>>> table[0]

[‘rabbit-hole’, ”, ‘B+’, ‘B’, ”, ‘C’, ‘C+’, ”, ‘F’, ‘B+’, ‘F’, ‘C’, ‘F’, ‘D’,
”, ”, ‘A’, ”, ”, ”, ”, ‘B+’, ‘C+’, ”, ”, ”, ”, ”, ”, ‘C+’, ”, ”,
”, ”, ”, ”, ‘A’, ”, ”, ”, ”, ”, ‘A’, ”, ”, ‘B+’, ‘B+’, ‘B’, ”, ”,
”, ‘D’, ‘B+’, ”, ”, ‘C+’, ”, ”, ”, ”, ”, ”, ‘B+’, ”, ”, ”, ”, ”,
”, ‘A’, ”, ”, ”, ”, ”, ”, ”, ‘D’, ”, ”,’C+’, ‘A’, ”, ”, ”, ‘C+’, ”]

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