MATH 20: PROBABILITY Midterm 1 Xingru Chen - PowerPoint PPT Presentation

MATH 20: PROBABILITY Midterm 1 Xingru Chen xingru.chen.gr@dartmouth.edu XC 2020

Ex Exam How many hours you spend preparing for the exam? Wrapper Wr How many hours you spend on the exam? Which problem you enjoy most? for midterm 1 What kind of problems would you suggest next time? … XC 2020

Problem 1: True or False XC 2020

Density Functions of Continuous Random Variable § The probability of occurrence of an event of the form [𝑦, 𝑦 + 𝑒𝑦] , where 𝑒𝑦 is small, can be estimated by 𝒛 𝑄 [𝑦, 𝑦 + 𝑒𝑦] ≈ 𝑔 𝑦 𝑒𝑦 . § As 𝑒𝑦 → 0 , the above probability approaches 0 , so that the probability of a single point 𝑦, 𝑄({𝑦}) is 0 . 𝑦 𝑦 + 𝑒𝑦 𝒚 XC 2020

Problem 1: True or False XC 2020

Problem 3: Proof XC 2020

Binomial Theorem = ! ! (𝑏 + 𝑐) ! = ∑ "#$ " 𝑏 " 𝑐 !%" . Let 𝑏 = 𝑐 = 1 , we have = 2 ! = ! ! ! ! ! . $ + & + ' + ⋯ + Let 𝑏 = −1 , 𝑐 = 1 , we have = ! ! ! ' − ⋯ + (−1) ! ! ! . 0 = $ − & + XC 2020

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Problem 4: Manipulation XC 2020

! 0 ≤ 𝑎 ≤ 1 XC 2020

! 𝑎 > 1 XC 2020

Problem 4: Manipulation XC 2020

0 ≤ |𝑌 − 𝑍| ≤ 1 ! 2 0 ≤ 𝑎 ≤ 1 ! 4 XC 2020

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Problem 5: National Committee of Senators XC 2020

! 𝑄 𝐹 = 1 − 𝑄(𝐹 ( ) XC 2020

Problem 6: Star Trek: Long and Prosper XC 2020

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Problem 6: Star Trek: Long and Prosper XC 2020

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Problem 7: Role Playing Game (RPG) XC 2020

Problem 7: Role Playing Game (RPG) XC 2020

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An interview question by video game companies XC 2020

Problem 7: Role Playing Game (RPG) 1 → 0 4 → 5 XC 2020

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DICE ROLLS IN ROLE PLAYING GAMES As a mathematician, you can find a job in a video game company! XC 2020

RPGs use some sort of randomizer when resolving actions. § Most often dice are used for this, but a few games use cards, rock- paper-scissors or other means of randomization. § There are dozens of different ways dice have been used in RPGs, and we are likely to see many more in the future. § This is not an evolution from bad methods to better methods. There is no such thing as a perfect dice-roll system suitable for all games. § How will a designer be able to decide which of the existing dice-roll method is best suited for his or her game, or when to invent his or her own? § It is in many ways an art. But like any art, there is an element of craft involved. XC 2020

Problem 8: Role Playing Game (continued) XC 2020

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Prior Pr probab ability Posterior Po probability lity of lity represents The pr prior pr probabili an event (often simply The po post ster erior pr probabili called th rior ) is its probability obtained from the up lity after taking into the pri updated pr prior pr probabili some prior information. account some new piece of information. Evidence ce Like kelihood ood ce term in kelihood represents The ev eviden ence Bayes’ theorem refers to The like a conditional the ov overall pr probabili lity of this new piece of probability. It is the degree to which the first information. event is consistent with the second event. Posterior probability = )*+,* -*,./.+0+12×4+560+7,,8 . B B 9:+86;<6 XC 2020

Po Posterior probability Prior Pr probab ability lity represents The po post ster erior pr probabili lity of The pr prior pr probabili an event (often simply the up lity after taking into updated pr prior pr probabili called th rior ) is its probability obtained from the pri account some new piece of information. some prior information. 𝑆 | ⚔⚔⚔⚔ in a row 𝑇𝑆 | ⚔⚔⚔⚔ in a row or 𝑆 , 𝑇𝑆 , or 𝑇𝑇𝑆 𝑇𝑇𝑆 | ⚔⚔⚔⚔ in a row Like kelihood ood Evidence ce The like kelihood represents a conditional The ev eviden ence ce term in Bayes’ theorem refers to probability. It is the degree to which the first the ov overall pr probabili lity of this new piece of event is consistent with the second event. information. ⚔⚔⚔⚔ in a row | 𝑆 ⚔⚔⚔⚔ in a row | 𝑇𝑆 or ⚔⚔⚔⚔ in a row ⚔⚔⚔⚔ in a row | 𝑇𝑇𝑆 XC 2020

Ba Bayes’ es’ f formula 𝑄 𝐹 ∩ 𝐼 ! = 𝑄 𝐹 𝐼 ! 𝑄(𝐼 ! ) . 𝑄 𝐼 = 𝐹 = 𝑄 𝐼 = 𝑄(𝐹|𝐼 = ) = 𝑄(𝐹) > 𝑄(𝐹 ∩ 𝐼 = ) . > 𝑄 𝐹 𝐼 = 𝑄(𝐼 = ) . 𝑄 𝐹 = ∑ =#& 𝑄 𝐹 = ∑ =#& Ba Bayes’ es’ f formula 𝑄 𝐼 = 𝑄(𝐹|𝐼 = ) 𝑄 𝐼 = 𝐹 = > 𝑄 𝐹 𝐼 = 𝑄(𝐼 = ) ∑ =#& XC 2020

Evidence ce The ev eviden ence ce term in Bayes’ theorem refers to the ov overall pr probabili lity of this new piece of information. ⚔⚔⚔⚔ in a row Evidence 𝑄 ⚔⚔⚔⚔ in a row = 𝑄 ⚔⚔⚔⚔ in a row | 𝑆 𝑄 𝑆 + = 𝑄 ⚔⚔⚔⚔ in a row | 𝑇𝑆 𝑄 𝑇𝑆 + 𝑄 ⚔⚔⚔⚔ in a row | 𝑇𝑇𝑆 𝑄 𝑇𝑇𝑆 XC 2020

2 or 3 Steps prior probability 01 𝑆 , 𝑇𝑆 , or 𝑇𝑇𝑆 posterior probability 02 𝑆 | ⚔⚔⚔⚔ in a row 𝑇𝑆 | ⚔⚔⚔⚔ in a row or 𝑇𝑇𝑆 | ⚔⚔⚔⚔ in a row new probability 03 ⚔ in the next mission XC 2020

Posterior Po probability Ne New prob obability 𝑆 | ⚔⚔⚔⚔ in a row 𝑇𝑆 | ⚔⚔⚔⚔ in a row or ⚔ in the next mission 𝑇𝑇𝑆 | ⚔⚔⚔⚔ in a row New probability 𝑄 ⚔ next = 𝑄 ⚔ next | 𝑆 𝑄 𝑆 + = 𝑄 ⚔ next | 𝑇𝑆 𝑄 𝑇𝑆 + 𝑄 ⚔ next | 𝑇𝑇𝑆 𝑄 𝑇𝑇𝑆 XC 2020

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Problem 9: A Random Walk Down Wall Street XC 2020

𝑣𝑇 𝑢 , with prob 𝑞 𝑇 𝑢 + 1 = = 𝑒𝑇(𝑢), with prob 1 − 𝑞 XC 2020

Problem 9: A Random Walk Down Wall Street XC 2020

Jo Jonathan Lee v Experiment with both smaller and longer periods of time. Try incorporating machine learning to assist with pattern recognition. Incorporate Bayesian probability. Stock market trends and people’s decisions to buy/sell are based on peoples’ beliefs. XC 2020

Sa Samuel l Bake ker v Switch the model from bigram to trigram. Bayesian learning model and Bayesian regression. … XC 2020

Ga Gaye yeong Son Song v XC 2020

Ga Gaye yeong Son Song v XC 2020

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