18.600: Lecture 39 Review: practice problems
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1 18.600: Lecture 39 Review: practice problems Scott Sheffield MIT
2 Markov chains Alice and Bob share a home with a bathroom, a walk-in closet, and 2 towels.
3 Markov chains Alice and Bob share a home with a bathroom, a walk-in closet, and 2 towels. Each morning a fair coin decide which of the two showers first.
4 Markov chains Alice and Bob share a home with a bathroom, a walk-in closet, and 2 towels. Each morning a fair coin decide which of the two showers first. After Bob showers, if there is at least one towel in the bathroom, Bob uses the towel and leaves it draped over a chair in the walk-in closet. If there is no towel in the bathroom, Bob grumpily goes to the walk-in closet, dries off there, and leaves the towel in the walk-in closet
5 Markov chains Alice and Bob share a home with a bathroom, a walk-in closet, and 2 towels. Each morning a fair coin decide which of the two showers first. After Bob showers, if there is at least one towel in the bathroom, Bob uses the towel and leaves it draped over a chair in the walk-in closet. If there is no towel in the bathroom, Bob grumpily goes to the walk-in closet, dries off there, and leaves the towel in the walk-in closet When Alice showers, she first checks to see if at least one towel is present. If a towel is present, she dries off with that towel and returns it to the bathroom towel rack. Otherwise, she cheerfully retrieves both towels from the walk-in closet, then showers, dries off and leaves both towels on the rack.
6 Markov chains Alice and Bob share a home with a bathroom, a walk-in closet, and 2 towels. Each morning a fair coin decide which of the two showers first. After Bob showers, if there is at least one towel in the bathroom, Bob uses the towel and leaves it draped over a chair in the walk-in closet. If there is no towel in the bathroom, Bob grumpily goes to the walk-in closet, dries off there, and leaves the towel in the walk-in closet When Alice showers, she first checks to see if at least one towel is present. If a towel is present, she dries off with that towel and returns it to the bathroom towel rack. Otherwise, she cheerfully retrieves both towels from the walk-in closet, then showers, dries off and leaves both towels on the rack. Problem: describe towel-distribution evolution as a Markov chain and determine (over the long term) on what fraction of days Bob emerges from the shower to find no towel.
7 Markov chains answers Let state 0, 1, 2 denote bathroom towel number.
8 Markov chains answers Let state 0, 1, 2 denote bathroom towel number. Shower state change Bob: 2 1, 1 0, 0 0.
9 Markov chains answers Let state 0, 1, 2 denote bathroom towel number. Shower state change Bob: 2 1, 1 0, 0 0. Shower state change Alice: 2 2, 1 1, 0 2.
10 Markov chains answers Let state 0, 1, 2 denote bathroom towel number. Shower state change Bob: 2 1, 1 0, 0 0. Shower state change Alice: 2 2, 1 1, 0 2. Morning state change AB: 2 1, 1 0, 0 1.
11 Markov chains answers Let state 0, 1, 2 denote bathroom towel number. Shower state change Bob: 2 1, 1 0, 0 0. Shower state change Alice: 2 2, 1 1, 0 2. Morning state change AB: 2 1, 1 0, 0 1. Morning state change BA: 2 1, 1 2, 0 2.
12 Markov chains answers Let state 0, 1, 2 denote bathroom towel number. Shower state change Bob: 2 1, 1 0, 0 0. Shower state change Alice: 2 2, 1 1, 0 2. Morning state change AB: 2 1, 1 0, 0 1. Morning state change BA: 2 1, 1 2, 0 2. Markov chain matrix: M =
13 Markov chains answers Let state 0, 1, 2 denote bathroom towel number. Shower state change Bob: 2 1, 1 0, 0 0. Shower state change Alice: 2 2, 1 1, 0 2. Morning state change AB: 2 1, 1 0, 0 1. Morning state change BA: 2 1, 1 2, 0 2. Markov chain matrix: M = Row vector π such that πm = π (with components of π summing to one) is ( ).
14 Markov chains answers Let state 0, 1, 2 denote bathroom towel number. Shower state change Bob: 2 1, 1 0, 0 0. Shower state change Alice: 2 2, 1 1, 0 2. Morning state change AB: 2 1, 1 0, 0 1. Morning state change BA: 2 1, 1 2, 0 2. Markov chain matrix: M = Row vector π such that πm = π (with components of π summing to one) is ( ). Bob finds no towel only if morning starts in state zero and Bob goes first. Over long term Bob finds no towel = 1 9 fraction of the time.
15 Optional stopping, martingales, central limit theorem Suppose that X 1, X 2, X 3,... is an infinite sequence of independent random variables which are each equal to 1 with probability 1/2 and 1 with probability 1/2. Let Y n = n i=1 X i. Answer the following: What is the the probability that Y n reaches 25 before the first time that it reaches 5?
16 Optional stopping, martingales, central limit theorem Suppose that X 1, X 2, X 3,... is an infinite sequence of independent random variables which are each equal to 1 with probability 1/2 and 1 with probability 1/2. Let Y n = n i=1 X i. Answer the following: What is the the probability that Y n reaches 25 before the first time that it reaches 5? Use the central limit theorem to approximate the probability that Y is greater than 6000.
17 Optional stopping, martingales, central limit theorem answers p p 5 5 = 0 and p 25 + p 5 = 1. Solving, we obtain p 25 = 1/6 and p 5 = 5/6.
18 Optional stopping, martingales, central limit theorem answers p p 5 5 = 0 and p 25 + p 5 = 1. Solving, we obtain p 25 = 1/6 and p 5 = 5/6. One standard deviation is = We want probability to be 2 standard deviations above mean. Should be about 1 2 2π e x2 /2 dx.
19 Martingales Let X i be independent random variables with mean zero. In which of the cases below is the sequence Y i necessarily a martingale?
20 Martingales Let X i be independent random variables with mean zero. In which of the cases below is the sequence Y i necessarily a martingale? Yn = n i=1 ix i
21 Martingales Let X i be independent random variables with mean zero. In which of the cases below is the sequence Y i necessarily a martingale? Yn = n i=1 ix i Y n = n i=1 X i 2 n
22 Martingales Let X i be independent random variables with mean zero. In which of the cases below is the sequence Y i necessarily a martingale? Yn = n i=1 ix i Y n = n i=1 X i 2 n Y n = n i=1 (1 + X i)
23 Martingales Let X i be independent random variables with mean zero. In which of the cases below is the sequence Y i necessarily a martingale? Yn = n i=1 ix i Y n = n i=1 X i 2 n Y n = n i=1 (1 + X i) Yn = n i=1 (X i 1)
24 Martingales Yes, no, yes, no.
25 Calculations like those needed for Black-Scholes derivation Let X be a normal random variable with mean 0 and variance 1. Compute the following (you may use the function Φ(a) := a 1 2π e x2 /2 dx in your answers):
26 Calculations like those needed for Black-Scholes derivation Let X be a normal random variable with mean 0 and variance 1. Compute the following (you may use the function Φ(a) := a 1 2π e x2 /2 dx in your answers): E[e 3X 3 ].
27 Calculations like those needed for Black-Scholes derivation Let X be a normal random variable with mean 0 and variance 1. Compute the following (you may use the function Φ(a) := a 1 2π e x2 /2 dx in your answers): E[e 3X 3 ]. E[e X 1 X (a,b) ] for fixed constants a < b.
28 Calculations like those needed for Black-Scholes derivation answers E[e 3X 3 ] = = = = e 3/2 = e 3/2 e 3x 3 1 2π e x2 /2 dx 1 2π e x2 6x+6 2 dx 1 2π e x2 6x+9 2 e 3/2 dx 1 2π e (x 3)2 2 dx
29 Calculations like those needed for Black-Scholes derivation answers E[e X 1 X (a,b) ] = = = b a b a b a = e 1/2 b e x 1 2π e x2 /2 dx e x 1 2π e x2 2 dx 1 2π e x2 2x dx a = e 1/2 b 1 a 1 1 2π e (x 1)2 2 dx 1 2π e x2 2 dx = e 1/2 (Φ(b 1) Φ(a 1))
30 If you want more probability and statistics... UNDERGRADUATE: (a) Introduction to Stochastic Processes (b) Topics in Math with Applications in Finance (c) Statistics for Applications
31 If you want more probability and statistics... UNDERGRADUATE: (a) Introduction to Stochastic Processes (b) Topics in Math with Applications in Finance (c) Statistics for Applications GRADUATE LEVEL PROBABILITY (a) Theory of Probability (b) Stochastic calculus (c) Topics in stochastic processes (topics vary repeatable, offered twice next year)
32 If you want more probability and statistics... UNDERGRADUATE: (a) Introduction to Stochastic Processes (b) Topics in Math with Applications in Finance (c) Statistics for Applications GRADUATE LEVEL PROBABILITY (a) Theory of Probability (b) Stochastic calculus (c) Topics in stochastic processes (topics vary repeatable, offered twice next year) GRADUATE LEVEL STATISTICS (a) Mathematical statistics (b) Topics in statistics (topics vary topic this year was machine learning; repeatable)
33 If you want more probability and statistics... UNDERGRADUATE: (a) Introduction to Stochastic Processes (b) Topics in Math with Applications in Finance (c) Statistics for Applications GRADUATE LEVEL PROBABILITY (a) Theory of Probability (b) Stochastic calculus (c) Topics in stochastic processes (topics vary repeatable, offered twice next year) GRADUATE LEVEL STATISTICS (a) Mathematical statistics (b) Topics in statistics (topics vary topic this year was machine learning; repeatable) OUTSIDE OF MATH DEPARTMENT (a) Look up new MIT minor in statistics and data sciences. (b) Look up long list of probability/statistics courses (about 78 total) at (c) Ask other MIT faculty how they use probability and statistics in their research.
34 Thanks for taking the course! Considering previous generations of mathematically inclined MIT students, and adopting a frequentist point of view...
35 Thanks for taking the course! Considering previous generations of mathematically inclined MIT students, and adopting a frequentist point of view... You will probably do some important things with your lives.
36 Thanks for taking the course! Considering previous generations of mathematically inclined MIT students, and adopting a frequentist point of view... You will probably do some important things with your lives. I hope your probabilistic shrewdness serves you well.
37 Thanks for taking the course! Considering previous generations of mathematically inclined MIT students, and adopting a frequentist point of view... You will probably do some important things with your lives. I hope your probabilistic shrewdness serves you well. Thinking more short term...
38 Thanks for taking the course! Considering previous generations of mathematically inclined MIT students, and adopting a frequentist point of view... You will probably do some important things with your lives. I hope your probabilistic shrewdness serves you well. Thinking more short term... Happy exam day!
39 Thanks for taking the course! Considering previous generations of mathematically inclined MIT students, and adopting a frequentist point of view... You will probably do some important things with your lives. I hope your probabilistic shrewdness serves you well. Thinking more short term... Happy exam day! And may the odds be ever in your favor.
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