Math 381 Discrete Mathematical Modeling

Transcription

1 Math 381 Discrete Mathematical Modeling Sean Griffin Today: -Projects -Central Limit Theorem -Markov Chains

2 Handout Projects Deadlines: Project groups and project descriptions due w/ homework (Due 7/23) Project Proposal (Due 7/25) Written Portion (Due 8/13) Poster Session (Due 8/17) Poster session poll

3 Project Proposal Abstract State the problem There should be a very clear goal in mind, and it should be mathematical in nature. It should be clear that there are ways to improve the model that are within reach. Start small, then start adding complexity to your model!! Importance of problem It must be a problem that you care about, and will care about finding a solution!! It doesn t have to be operations research.

4 Methods Project Proposal Preferably methods related to what we have talked about or will talk about in class. -Linear Programming -Graph Theory -Statistical analysis -Markov Chains -Fair divisions If your whole group is comfortable with outside material, then we can try to work something out.

5 Project Proposal Data Should use real data whenever possible. Be skeptical of the data you use. -Is it from a reliable source? You will need to cite this in your work!! -Is it biased? -Is it a complete set of data, or a small sample?

6 Note: If you don t satisfactorily answer some of these questions in your proposal, I may make your group revise and resubmit. I will be gone next week but reachable by . If you need help finding a group, please let me know and I ll help you find a group. I will be back the following week July I encourage all groups to meet with me that week. I would like to meet will all groups the week of August 6-10.

7 Linear Programming Diet Problems Transportation Project Ideas Puzzles/Games -Tantrix, Sudoku, Battleship etc. Sports - Fantasy teams, Resting players,etc.

8 Graph Theory Graph coloring -Sports scheduling -Seating plans Shortest path Clustering Network Flows Markov Chains Autocorrect Games -RISK, blackjack, Monopoly, etc. Music -Auto compose music

9 Back to Stochastic Processes

10 Sample Mean Now suppose we throw our fair die 50 times and record the mean of this data set. What number should this be close to? Exactly! It should be close to the expected value E(Y ). Now suppose we do the following: Throw the die 50 more times and record the mean. Call that one run. Then throw it 50 more times and record the mean. That s another run. Keep going. Do 1000 runs, say. At the end, we ll have a data set consisting of 100 sample means. Let s plot a histogram of this data.

11 1000 runs of 50 die rolls

12 Question What do you think the histogram will look like if we used the unfair die from before, where 5 and 6 were more likely?

13 Rolling an Unfair Die Unfair Die y p(y) 1/24 1/24 1/24 1/24 1/2 1/3

14 Rolling an Unfair Die Unfair Die y p(y) 1/24 1/24 1/24 1/24 1/2 1/3 One way to sample from this distribution: 1) Use a pseudorandom number generator to pick a number between 1 and 24 (the common denominator). 2) Split the range {1, 2,..., 24} into six bins

15 Rolling an Unfair Die Unfair Die y p(y) 1/24 1/24 1/24 1/24 1/2 1/3 One way to sample from this distribution: 1) Use a pseudorandom number generator to pick a number between 1 and 24 (the common denominator). 2) Split the range {1, 2,..., 24} into six bins

16 Rolling an Unfair Die Unfair Die y p(y) 1/24 1/24 1/24 1/24 1/2 1/3 One way to sample from this distribution: 1) Use a pseudorandom number generator to pick a number between 1 and 24 (the common denominator). 2) Split the range {1, 2,..., 24} into six bins ) If the random number generator outputs a number j which is in bin i, then consider it the same as rolling an i with the unfair die.

17 Rolling an Unfair Die If the probabilities have a large denominator, or if they are irrational: Choose a large number of bins, and then deal with the fractional excess seperately. Accept-reject method -Instead sample uniformly at random on the plane and accept the samples which lie in the region under the graph of the density function. - Throwing darts on a dart board Use a continous uniform random variable X (0, 1) and chop up the interval into the given lengths.

18 Simulating Unfair Die

19 Normal Distribution The histogram of means we calculated is in the shape of a bell curve. The data is normally distributed. Probability density of the normal distribution: f(x) = 1 2πσ 2 (x µ)2 e 2σ 2 where µ is the mean and σ 2 is the variance (here, σ is the standard deviation).

20 Central Limit Theorem Let Y 1, Y 2,..., Y n be independent and identically distributed (i.i.d.) random variables.

21 Central Limit Theorem Let Y 1, Y 2,..., Y n be independent and identically distributed (i.i.d.) random variables. In our example, Y i is the outcome of the ith roll of the die. Each of these random variables are independent.

22 Central Limit Theorem Let Y 1, Y 2,..., Y n be independent and identically distributed (i.i.d.) random variables. In our example, Y i is the outcome of the ith roll of the die. Each of these random variables are independent. What can we say about the distribution of the sample average, Y = (Y Y n )/n

23 Central Limit Theorem Let Y 1, Y 2,..., Y n be independent and identically distributed (i.i.d.) random variables. In our example, Y i is the outcome of the ith roll of the die. Each of these random variables are independent. What can we say about the distribution of the sample average, Y = (Y Y n )/n Theorem 1 (Central Limit Theorem). The sample average Y tends to a normal distribution as the sample size tends to infinity.

24 Central Limit Theorem Let Y 1, Y 2,..., Y n be independent and identically distributed (i.i.d.) random variables. In our example, Y i is the outcome of the ith roll of the die. Each of these random variables are independent. What can we say about the distribution of the sample average, Y = (Y Y n )/n Theorem 1 (Central Limit Theorem). The sample average Y tends to a normal distribution as the sample size tends to infinity. It doesn t matter how the Y i are distributed, as long as they are sampled from the same distribution. In other words, the same thing will happen if we use an unfair (weighted) die.

25 Rolling an Unfair Die Let X i have the following probability density function: y p(y) 1/6 1/6 2/3 Bar graph for the probability density function of: X 1

26 Rolling an Unfair Die Let X i have the following probability density function: y p(y) 1/6 1/6 2/3 Bar graph for the probability density function of: X 1 + X 2

27 Rolling an Unfair Die Let X i have the following probability density function: y p(y) 1/6 1/6 2/3 Bar graph for the probability density function of: X 1 + X 2 + X 3

28 Rolling an Unfair Die Let X i have the following probability density function: y p(y) 1/6 1/6 2/3 Bar graph for the probability density function of: X 1 + X 2 + X 3 + X 4

29 Rolling an Unfair Die Let X i have the following probability density function: y p(y) 1/6 1/6 2/3 Bar graph for the probability density function of: X 1 + X 2 + X 3 + X 4 + X 5

30 Rolling an Unfair Die Let X i have the following probability density function: y p(y) 1/6 1/6 2/3 Bar graph for the probability density function of: X 1 + X 2 + X 3 + X 4 + X 5 + X 6

31 Rolling an Unfair Die Let X i have the following probability density function: y p(y) 1/6 1/6 2/3 Bar graph for the probability density function of: X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + X 8 + X 9

32 Continuous Random Variable A random variable, X, is a function that associated a number with each point in an experiment s sample space We say X is a continuous random variable if there is an interval of positive length such that X takes on all values in the interval. Example The amount of time it takes to recover from the flu. Modeling the spread of disease is a fun project!!

33 Properties of Normal Distribution We say that X has normal distribution if for some µ and σ > 0, its density function is 1 f(x) = 2πσ 2 (x µ)2 e 2σ 2 In this case, we say X is N(µ, σ 2 ). If X is N(µ, σ 2 ), then cx is N(cµ, c 2 σ 2 ). If X is N(µ, σ 2 ), then X + c is N(µ + c, σ 2 ). If X 1 is N(µ 1, σ 2 1), and X 1 and X 2 are independent, then X 1 + X 2 is N(µ 1 + µ 2, σ σ 2 2).

34 Markov Chains

35 Markov Chain Suppose you had discrete random variables X 0, X 1, X 2...,. Then a discrete-time stochastic process is simply a description of the relation between the random variables. A Markov Chain is a discrete-time stochastic process which is memoryless : The dependence of X i+1 on X i is the same as the dependence of X i+2 on X i+1.

36 Gambler s Ruin Suppose a gambler starts at t = 0 with $2. At times 1, 2,..., he plays a game in which he bets $1. With probability p, he wins and gains $1. With probability 1 p, he loses $1. Suppose the game ends when he reaches $4 or $0. Let X t to be how much money the gambler has at time t. If he has $2 at time t, then with probability p he will have $3 at time t + 1.

37 Transition Matrix We can represent the situation at any time t with the following matrix: Next State $0 $1 $2 $3 $4 $ $1 1 p 0 p 0 0 Current State $2 0 1 p 0 p 0 $ p 0 p $ Call this matrix P = [p i,j ] i,j. P is the transition matrix of the Markov Chain.

38 Weighted Graph Representation It can also be represented with the following weighted graph: p p p 1 $0 $1 $2 $3 $4 1 p 1 p 1 p 1 Verticex set is the set of states Each arc (i, j) weighted by the transition probability p i,j.

39 n-step Transition Probabilites If the gambler starts with $i, then after 2 games what is the probability the gambler will have $j dollars?

40 n-step Transition Probabilites If the gambler starts with $i, then after 2 games what is the probability the gambler will have $j dollars? p 0 p p 0 p p 0 p Transition from t = 0 to t = p 0 p p 0 p p 0 p Transition from t = 1 to t = 2

41 n-step Transition Probabilites If the gambler starts with $i, then after 2 games what is the probability the gambler will have $j dollars? p 0 p p 0 p p 0 p Transition from t = 0 to t = 1 = p 0 p p 0 p p 0 p Transition from t = 1 to t = p p(1 p) 0 p 2 0 (p 1) 2 0 2p(1 p) 0 p 2 0 (p 1) 2 0 p(1 p) p = P Transition from t = 0 to t = 2

42 n-step Transition Probabilites If the gambler starts with $i, then after 2 games what is the probability the gambler will have $j dollars? It is the (i, j) entry of P 2. Similarly, after n games his probability of having $j is the (i, j) entry of P n. We call P n the n-step transition matrix.

43 Midterm Exam Next Time

44 References Normal Distribution