The Evolution of Random Phenomena

Transcription

1 The Evolution of Random Phenomena A Look at Markov Chains Glen Wang glenw@uchicago.edu Splash! Chicago: Winter Cascade 2012 Lecture 1: What is Randomness?

2 What is randomness? Can you think of some examples of randomness?

3 What is randomness? Can you think of some examples of randomness? Rolling dice. Flipping coins. Card games. Picking numbers out of a hat.

4 What is randomness? Can you think of some examples of randomness? Rolling dice. Flipping coins. Card games. Picking numbers out of a hat. We use the word random in daily conversation.

5 What is randomness? Can you think of some examples of randomness? Rolling dice. Flipping coins. Card games. Picking numbers out of a hat. We use the word random in daily conversation. We just randomly met on the street and... Well, that was random...

6 What is randomness? Can you think of some examples of randomness? Rolling dice. Flipping coins. Card games. Picking numbers out of a hat. We use the word random in daily conversation. We just randomly met on the street and... Well, that was random... How much do we really know about randomness?

7 Let s do an experiment! In the beginning of class, I gave you a strip of paper with the numbers 1 to 20 on it. Randomly circle 4 numbers on your strip of paper.

8 Computer-generated selections I asked a computer to pick 4 numbers randomly just like in our experiment. Here are the computer s results:

9 Computer-generated selections I asked a computer to pick 4 numbers randomly just like in our experiment. Here are the computer s results:

10 Computer-generated selections I asked a computer to pick 4 numbers randomly just like in our experiment. Here are the computer s results:

11 Computer-generated selections I asked a computer to pick 4 numbers randomly just like in our experiment. Here are the computer s results:

12 Computer-generated selections I asked a computer to pick 4 numbers randomly just like in our experiment. Here are the computer s results:

13 Computer-generated selections I asked a computer to pick 4 numbers randomly just like in our experiment. Here are the computer s results:

14 Computer-generated selections I asked a computer to pick 4 numbers randomly just like in our experiment. Here are the computer s results: How do these results compare with ours?

15 The goals of this lecture 1 Define randomness. 2 Develop a mathematical language to talk about randomness,

16 The goals of this lecture 1 Define randomness. 2 Develop a mathematical language to talk about randomness, which means, probability.

17 Defining randomness What is a process? Q: Recall our examples of randomness. What do all of these have in common?

18 Defining randomness What is a process? Q: Recall our examples of randomness. What do all of these have in common? A: They are all actions and have outcomes. e.g. Rolling a die is an action, and outcomes could be any number from 1 to 6. e.g. Flipping a coin is an action, with heads or tails as outcomes.

19 Defining randomness What is a process? Q: Recall our examples of randomness. What do all of these have in common? A: They are all actions and have outcomes. e.g. Rolling a die is an action, and outcomes could be any number from 1 to 6. e.g. Flipping a coin is an action, with heads or tails as outcomes. Definition A process is anything that responds to action to produce outcomes.

20 Defining randomness What is a process? Q: Recall our examples of randomness. What do all of these have in common? A: They are all actions and have outcomes. e.g. Rolling a die is an action, and outcomes could be any number from 1 to 6. e.g. Flipping a coin is an action, with heads or tails as outcomes. Definition A process is anything that responds to action to produce outcomes. We can think of this as cause and effect, stimulus and response, input and output.

21 Defining randomness Function as a process

22 Defining randomness Function as a process What are some examples of functions?

23 Defining randomness Function as a process What are some examples of functions? How can we tell if something is a function?

24 Defining randomness Function as a process What are some examples of functions? How can we tell if something is a function? Vertical line test.

25 Defining randomness Function as a process What are some examples of functions? How can we tell if something is a function? Vertical line test. We can think of a function as a process. The action is what we give the function as input, and the outcome is the function s output.

26 Defining randomness Function as a process What are some examples of functions? How can we tell if something is a function? Vertical line test. We can think of a function as a process. The action is what we give the function as input, and the outcome is the function s output. Definition A function is a process that for every input, gives a unique (only one) output.

27 Defining randomness A traffic example Inputs Red Outputs Stop Green Go

28 Defining randomness A traffic example Inputs Red Outputs Stop Yellow Green Go

29 Defining randomness A traffic example Inputs Outputs Stop Yellow Go

30 Defining randomness We arrive at a definition! Definition A random variable is a process that associates inputs to possibly more than one output.

31 Defining randomness We arrive at a definition! Definition A random variable is a process that associates inputs to possibly more than one output. The definition of random variable captures what we know so far about randomness.

32 Defining randomness We arrive at a definition! Definition A random variable is a process that associates inputs to possibly more than one output. The definition of random variable captures what we know so far about randomness. Q: What can we do to better understand the behavior of random variables?

33 Defining randomness We arrive at a definition! Definition A random variable is a process that associates inputs to possibly more than one output. The definition of random variable captures what we know so far about randomness. Q: What can we do to better understand the behavior of random variables? e.g. Are cars more likely to stop when facing a yellow light, or are they more likely to go?

34 A: Math!

35 A: Math!... in particular, probability!

36 What is probability? What comes to mind when we say probability?

37 What is probability? What comes to mind when we say probability? A measurement of likelihood. How random phenomena are expected to behave.

38 What is probability? What comes to mind when we say probability? A measurement of likelihood. How random phenomena are expected to behave. Let s motivate the definition through an example!

39 We roll a die... Suppose we roll a fair, six-sided die. What could happen?

40 We roll a die... Suppose we roll a fair, six-sided die. What could happen? A: We get one of the following results:

41 We roll a die... Suppose we roll a fair, six-sided die. What could happen? A: We get one of the following results: So rolling a die is a random variable where the input (action) is the roll, and the possible outputs are 1 to 6.

42 Outcomes vs. events Definition An outcome of a random variable is one of the possible outputs of that random variable. So rolling a 1, rolling a 2, etc. are all outcomes of the rolling a die random variable.

43 Outcomes vs. events Definition An event is a collection of one or more outcomes.

44 Outcomes vs. events Definition An event is a collection of one or more outcomes. e.g. The event of rolling a number greater than 3.

45 Outcomes vs. events Definition An event is a collection of one or more outcomes. e.g. The event of rolling a number greater than 3. e.g. We make a distinction between the event of rolling a 1 and the outcome 1.

46 What is probability?

47 What is probability? Definition Given a random variable, a probability is a way to assign numbers to all possible events that could occur.

48 What is probability? Definition Given a random variable, a probability is a way to assign numbers to all possible events that could occur. The way that a probability assigns numbers follows specific rules:

49 What is probability? Definition Given a random variable, a probability is a way to assign numbers to all possible events that could occur. The way that a probability assigns numbers follows specific rules: The numbers assigned must be between 0 and 1.

50 What is probability? Definition Given a random variable, a probability is a way to assign numbers to all possible events that could occur. The way that a probability assigns numbers follows specific rules: The numbers assigned must be between 0 and 1. An event assigned with probability 0 will never occur.

51 What is probability? Definition Given a random variable, a probability is a way to assign numbers to all possible events that could occur. The way that a probability assigns numbers follows specific rules: The numbers assigned must be between 0 and 1. An event assigned with probability 0 will never occur. An event assigned with probability 1 will always occur.

52 What is probability? Definition Given a random variable, a probability is a way to assign numbers to all possible events that could occur. The way that a probability assigns numbers follows specific rules: The numbers assigned must be between 0 and 1. An event assigned with probability 0 will never occur. An event assigned with probability 1 will always occur. If every outcome in event A is also in event B, then P(A) P(B).

53 What is probability? Definition Given a random variable, a probability is a way to assign numbers to all possible events that could occur. The way that a probability assigns numbers follows specific rules: The numbers assigned must be between 0 and 1. An event assigned with probability 0 will never occur. An event assigned with probability 1 will always occur. If every outcome in event A is also in event B, then P(A) P(B). If two events A, B have no outcomes in common, then the probability that an outcome from one of the two events occurs equals the sum of the probabilities assigned to each event: P(A or B) = P(A) + P(B).

54 What is probability? Probability is a function! This function is defined differently for different random variables. The inputs are events and the outputs are numbers between 0 and 1.

55 Law of total probability Theorem (Law of total probability) For any random variable, the probability that some outcome occurs is 1. Inputs Yellow Outputs Stop Go

56 A quick example of a Markov chain

57 A quick example of a Markov chain 0.7 Red Red 1 Stop 0.3 Green Green 1 Go 0.3 Red 0.7 Green

58 A quick example of a Markov chain 0.7 Red Red Green Red 0.3 Red Green Green Red Green Green Red Green Red Green

59 Summary A process is any mechanism that gives outputs (responses) to certain inputs (actions). A function is a process that assigns every input only one output. A random variable is a process that assigns every input with possibly many outputs. One output of a random variable is called an outcome. A collection of outcomes is called an event. Probability is a specific way to assign events to numbers between 0 and 1. The probability that some event will occur is 1 (law of total probability).

60 Note card questions! Please take a minute to fill out the following questions on your note card. Your input will let me know how the class is going for you, so in the future I can set up the class according to your needs. 1 Was the pacing of this class too fast? Too slow? Just right? 2 Is there anything that you didn t quite understand? If so, please tell me what it is. 3 What were some of the things you liked about this lecture? Why? If you have questions for me, I will be sticking around after class. Please feel free to ask me anything! You can also me at glenw@uchicago.edu.