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

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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?

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MATH 20: PROBABILITY

Midterm 1 Xingru Chen xingru.chen.gr@dartmouth.edu

XC 2020

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Ex Exam Wr Wrapper

for midterm 1

How many hours you spend preparing for the exam? How many hours you spend

  • n

the exam? Which problem you enjoy most? What kind

  • f

problems would you suggest next time? …

XC 2020

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Problem 1: True or False

XC 2020

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Density Functions of Continuous Random Variable

Β§ The probability

  • f
  • ccurrence
  • f

an event

  • f

the form [𝑦, 𝑦 + 𝑒𝑦], where 𝑒𝑦 is small, can be estimated by 𝑄 [𝑦, 𝑦 + 𝑒𝑦] β‰ˆ 𝑔 𝑦 𝑒𝑦. Β§ As 𝑒𝑦 β†’ 0, the above probability approaches 0, so that the probability

  • f

a single point 𝑦, 𝑄({𝑦}) is 0. 𝒛 π’š 𝑦 𝑦 + 𝑒𝑦

XC 2020

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Problem 1: True or False

XC 2020

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Problem 3: Proof

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=

Binomial Theorem (𝑏 + 𝑐)!= βˆ‘"#$

! ! " 𝑏"𝑐!%".

=

Let 𝑏 = 𝑐 = 1, we have

2! =

! $ + ! & + ! ' + β‹― + ! ! .

=

Let 𝑏 = βˆ’1, 𝑐 = 1, we have

0 =

! $ βˆ’ ! & + ! ' βˆ’ β‹― + (βˆ’1)! ! ! .

XC 2020

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SLIDE 8

XC 2020

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Problem 4: Manipulation

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! 0 ≀ π‘Ž ≀ 1

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! π‘Ž > 1

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Problem 4: Manipulation

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! 0 ≀ π‘Ž ≀ 1 4 ! 0 ≀ |π‘Œ βˆ’ 𝑍| ≀ 1 2

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Problem 5: National Committee of Senators

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!

𝑄 𝐹 = 1 βˆ’ 𝑄(𝐹()

XC 2020

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Problem 6: Star Trek: Long and Prosper

XC 2020

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Problem 6: Star Trek: Long and Prosper

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Problem 7: Role Playing Game (RPG)

XC 2020

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Problem 7: Role Playing Game (RPG)

XC 2020

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An interview question by video game companies

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Problem 7: Role Playing Game (RPG)

1 β†’ 0 4 β†’ 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

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RPGs use some sort

  • f

randomizer when resolving actions.

Β§ Most

  • ften

dice are used for this, but a few games use cards, rock- paper-scissors

  • r
  • ther

means

  • f

randomization. Β§ There are dozens

  • f

different ways dice have been used in RPGs, and we are likely to see many more in the future. Β§ 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. Β§ How will a designer be able to decide which

  • f

the existing dice-roll method is best suited for his

  • r

her game,

  • r

when to invent his

  • r

her

  • wn?

Β§ It is in many ways an

  • art. But

like any art, there is an element

  • f

craft involved.

XC 2020

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Problem 8: Role Playing Game (continued)

XC 2020

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Posterior probability = )*+,* -*,./.+0+12Γ—4+560+7,,8

9:+86;<6

.

B B

Pr Prior probab ability

The pr prior pr probabili lity of an event (often simply called th the pri rior) is its probability

  • btained

from some prior information.

Evidence ce

The ev eviden ence ce term in Bayes’ theorem refers to the ov

  • verall pr

probabili lity of this new piece

  • f

information.

Like kelihood

  • od

The like kelihood represents a conditional

  • probability. It

is the degree to which the first event is consistent with the second event.

Po Posterior probability

The po post ster erior pr probabili lity represents the up updated pr prior pr probabili lity after taking into account some new piece

  • f

information.

XC 2020

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Pr Prior probab ability

The pr prior pr probabili lity of an event (often simply called th the pri rior) is its probability

  • btained

from some prior information.

𝑆, 𝑇𝑆, or 𝑇𝑇𝑆 Evidence ce

The ev eviden ence ce term in Bayes’ theorem refers to the ov

  • verall pr

probabili lity of this new piece

  • f

information.

βš”βš”βš”βš” in a row Like kelihood

  • od

The like kelihood represents a conditional

  • probability. It

is the degree to which the first event is consistent with the second event.

βš”βš”βš”βš” in a row | 𝑆 βš”βš”βš”βš” in a row | 𝑇𝑆 or βš”βš”βš”βš” in a row | 𝑇𝑇𝑆 Po Posterior probability

The po post ster erior pr probabili lity represents the up updated pr prior pr probabili lity after taking into account some new piece

  • f

information.

𝑆 | βš”βš”βš”βš” in a row 𝑇𝑆 | βš”βš”βš”βš” in a row

  • r

𝑇𝑇𝑆 | βš”βš”βš”βš” in a row

XC 2020

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Ba Bayes’ es’ f formula

𝑄 𝐼= 𝐹 = 𝑄 𝐼= 𝑄(𝐹|𝐼=) 𝑄(𝐹) 𝑄 𝐹 = βˆ‘=#&

> 𝑄(𝐹 ∩ 𝐼=).

=

𝑄 𝐹 ∩ 𝐼! = 𝑄 𝐹 𝐼! 𝑄(𝐼!).

𝑄 𝐹 = βˆ‘=#&

> 𝑄 𝐹 𝐼= 𝑄(𝐼=).

Ba Bayes’ es’ f formula

𝑄 𝐼= 𝐹 = 𝑄 𝐼= 𝑄(𝐹|𝐼=) βˆ‘=#&

> 𝑄 𝐹 𝐼= 𝑄(𝐼=)

XC 2020

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=

Evidence 𝑄 βš”βš”βš”βš” in a row = 𝑄 βš”βš”βš”βš” in a row | 𝑆 𝑄 𝑆 + 𝑄 βš”βš”βš”βš” in a row | 𝑇𝑆 𝑄 𝑇𝑆 + 𝑄 βš”βš”βš”βš” in a row | 𝑇𝑇𝑆 𝑄 𝑇𝑇𝑆

Evidence ce

The ev eviden ence ce term in Bayes’ theorem refers to the ov

  • verall pr

probabili lity of this new piece

  • f

information.

βš”βš”βš”βš” in a row

XC 2020

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03 01 02

βš” in the next mission

new probability

𝑆, 𝑇𝑆, or 𝑇𝑇𝑆

prior probability

𝑆 | βš”βš”βš”βš” in a row 𝑇𝑆 | βš”βš”βš”βš” in a row

  • r

𝑇𝑇𝑆 | βš”βš”βš”βš” in a row

posterior probability

2 or 3 Steps

XC 2020

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=

New probability 𝑄 βš” next = 𝑄 βš” next | 𝑆 𝑄 𝑆 + 𝑄 βš” next | 𝑇𝑆 𝑄 𝑇𝑆 + 𝑄 βš” next | 𝑇𝑇𝑆 𝑄 𝑇𝑇𝑆

Ne New prob

  • bability

βš” in the next mission Po Posterior probability 𝑆 | βš”βš”βš”βš” in a row 𝑇𝑆 | βš”βš”βš”βš” in a row

  • r

𝑇𝑇𝑆 | βš”βš”βš”βš” in a row

XC 2020

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XC 2020

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Problem 9: A Random Walk Down Wall Street

XC 2020

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𝑇 𝑒 + 1 = = 𝑣𝑇 𝑒 , with prob π‘ž 𝑒𝑇(𝑒), with prob 1 βˆ’ π‘ž

XC 2020

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Problem 9: A Random Walk Down Wall Street

XC 2020

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v

Jo Jonathan Lee

Experiment with both smaller and longer periods

  • f

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

  • n

peoples’ beliefs.

XC 2020

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v

Sa Samuel l Bake ker

Switch the model from bigram to trigram. Bayesian learning model and Bayesian regression. …

XC 2020

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v

Ga Gaye yeong Son Song

XC 2020

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v

Ga Gaye yeong Son Song

XC 2020

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