2. Modeling Uncertainty

Transcription

1 2. Modeling Uncertainty Models for Uncertainty (Random Variables): Big Picture We now move from viewing the data to thinking about models that describe the data. Since the real world is uncertain, our models must incorporate uncertainty. Sometimes we have a pretty good idea about what these models should look like, sometimes they are approximations to complicated real world phenomenon.

2 What exactly is probability? Definition of probability: The probability of a given outcome (say a head on the coin toss) is defined as the fraction of times that we observe that outcome over a very very large number of realized outcomes (tosses). This is called a frequentist definition of probability..9.8 Fraction of times heads Number of tosses 2

3 6 4 2 What do you think the return will be next year? Smaller sample we are less certain What do you think the return will be next year?

4 Let s think about the basics: What do we need for a simple model of uncertainty? A model needs to specify all the possible outcomes. A model needs to tell us the probability of each of the uncertain outcomes. We also require that only one outcome can occur at a time (mutually exclusive assumptions) The technical term for models that satisfy these assumptions is RANDOM VARIABLE (RV). Consider the simple coin tossing example There are two possible outcomes, heads and tails. It is convenient to use numbers to enumerate the possible outcomes. Lets call the outcomes for heads and for tails. 4

5 Here is our model for a single coin toss Outcome Probability.5.5 There are two possible outcomes, heads and tails. Probabilities are given. All possible outcomes listed. Finally, lets give it a name. Lets call this model X. Don t get confused. The random variable is not data! It is a list of outcomes and probabilities (a table) that satisfy our RV assumptions. It is a model even though the notation is similar to the last section of notes. 5

6 Giving the random variable a name is convenient. Here s what we can do: We can refer to the probability that we get the outcome heads as Pr(X=). This is shorthand notation for go to the table and find the probability of a (head). Similarly, the probability that we get the outcome tails is Pr(X=) So we can quickly summarize the model by saying X= corresponds to the outcome heads, X= corresponds to the outcome tails. Pr(X=)=.5 and Pr(X=)=.5 Obviously, we can easily handle the case of more than just two outcomes. Alternatively,if we have the model X for the coin toss we can make statements about what can happen and how often events should occur over the long run: The two possible outcomes are equally likely. Over the long run, if we tossed the coin over and over, we expect a 5% of the time. Probability is the long run frequency. How often it happens 6

7 Equivalently (and this will be crucial later) we have: If we toss the coin n times with n really big then, if n is the number of s n is the number of s then, n n Pr X.5 Pr X.5 n n The coin tossing was convenient to start with because we knew the true probabilities. More generally, we are in the setting where we don t know the true probabilities. Consider a group of mortgages. Some of these homes will default on their mortgage. What is the probability that a given household will default? 7

8 Defaults Here is some data for a group of households Default We record if default if no default Fraction of default occurrence =.2 Consider a new mortgage, not in our data set. What will happen? Let Y denote the outcome no default () default () We could think of Y as an uncertain quantity with: Pr(Y )?? Pr(Y )?? 8

9 Unlike the coin example, it is not so obvious what to use for the probabilities. In our sample we have 2% defaults. Does that mean the probability of a default =.2? In the coin tossing example if we toss a coin times, do we expect to get 5 exactly heads for sure? We can think of the sample frequency as an estimate of the true probability. The larger the sample the closer the estimate should be to the true probability. Our model for defaults requires a probability of a default. Later in the course we will use past data to get an estimate of the probability. How good is the estimate? How much data do we need? 9

10 More generally we can have more than two possible outcomes: Suppose that we have 5 homes. Let x be the number of homes that default out of the 5. The possible outcomes are that none default, default, 2 up to all 5. Assuming the probability is the same for each home (.2) we know the model. Here is the model: This is also called the probability distribution for the number of homes that default

11 Notation We use various notations for the probability that the random variable X takes on value x: P(X x), Pr(x), p (x), p(x) X In each case the meaning is the same: the probability that the random variable X takes the value x. Don t get confused here. Little x denotes a possible numeric outcome for the random variable X, got it? For random variables, we add probabilities together to get the probability of any one of those outcomes occurring. Pr( a X b) p( x) a x b Careful, this rule only works for probabilities of mutually exclusive events!

12 Here is the distribution (again) for the number of mortgage defaults in a pool of 5 homes when the probability any single home defaults is.2 (you ll see where these come from later). Probability Density Function Binomial with n = 5 and p =.2 x P( X = x) What s the probability of no defaults? What s the probability of 2 or less defaults? What s the probability Of more than 6 defaults? What is the probability of,,2,,5 defaults? 2

13 Example Suppose you are considering investing in an asset. Let R denote the return next month. We think of R as a random variable!!! R: r.5..5 p(r)..5.4 The probability that the return is greater than.5 is.9. Graphing Discrete Random Variables We can use a graph to see the distribution of an RV. Simply plot x vs p(x):.5.4 p(r) r.5 3

14 Some types of random variables are so common that they have names (i.e. Normal or Gaussian distribution, Binomial, etc ) Let s begin with one of the most simple and common, the Bernoulli. The Bernoulli Distribution A Bernoulli random variable is one in which there are only two outcomes. We refer to these as outcome and outcome. Sometimes we call this a dummy variable. This could refer to a yes or no answer. A person s gender. Whether or not the stock market goes up or down tomorrow. Whether a person favors a candidate. The outcome of a coin toss etc 4

15 It s a whole family. We just need to give the probability of a (the probability of is ( Pr()). If X is a Bernoulli(p) then Outcome Probability ( p) p So our model for whether or not a household defaults is a Bernoulli model. We would say the probability of a house defaulting is Bernoulli(.2). This means: Default Probability

16 2.3 Joint models Many interesting modeling questions involve relationships between random outcomes. For example, consider the Home Affordable Refinance Program (HARP) Introduced in March 29, HARP enables borrowers with little or no equity to refinance into more affordable mortgages without new or additional mortgage insurance. HARP targets borrowers with loan to value (LTV) ratios equal to or greater than 8 percent and who have limited delinquencies over the 2 months prior to refinancing. Whether or not the program is extended is clearly related to the likelihood that a home defaults next year. Another example: there might be a relationship between bond price movements and stock price movements on the same day. If the bond market goes up tomorrow that might be related to what happens in the stock market. Clearly, a model for just the bond price moves and another model for just the stock price moves will not help us examine these types of relationships. We need something new. 6

17 How do we write down joint (bivariate) models for two variables? In the simple one variable case we wrote down the possible outcomes and the corresponding probabilities to get a model for the variable. We give the bivariate probability distribution of a pair of random variables by:. Listing out all the possible pairs (combinations) of values for both variables. 2. For each pair we give a probability. The sum of the probabilities over all pairs =. Notation: p(x,y) Pr(X x and Y y) The joint bivariate probabilities of X and Y is specified by the numbers pxy (, ) for all possible x and y. 7

18 Example Let X be the result of tossing a coin (=H, =T). Let Y be the result from a second coin. Then the joint distribution of X and Y is given by this table (x,y) p(x,y) (,).25 (,).25 (,).25 (,).25. We simply list out all possibilities for the pair and give each one a probability. A alternative way to display the joint model is: X Y We have a two way table where each spot in the table corresponds to a possible (x,y) pair. At each spot we give the prob of the corresponding pair 8

19 Example Let X and Y be returns on two different assets. What does this table say about the relationship between X and Y. What is the prob that they are equal? Y X 5% % 5% 5%..7.7 % % p p XY, XY, (. 5,. 5). (. 5,. ). 3 Example Let X and Y be returns on two different assets. What does this table say about the relationship between X and Y. What is the prob that they are equal? Y X 5% % 5% 5%..7.7 % % p p XY, XY, (. 5,. 5). (. 5,. ). 3 9

20 NB Probability still means the same thing! X We expect to see the pair (X,Y)=(.,.) 3% of the time. Y 5% % 5% % 5% % How do we go from the joint model back to the model for X and the model for Y? If we know the joint model of X and Y, we know how likely it is to see any given pair of x and y. From this, we should be able to figure out how likely we are to see an outcome for just x or just y, right? That is, if we know p XY (x,y) We ought to be able to figure out p X (x) and p Y (y) 2

21 Example X 5% % 5% Y 5% % 5% What is p X (.5)? p X (.5) p XY (.5,.5) p XY (.5,.) p XY (.5,.5) Example X 5% % 5% Y What is p X (.5)? 5% % 5% p X (.5) p XY (.5,.5) p XY (.5,.) p XY (.5,.5) 2

22 The Marginal Distributions Given the joint distribution of X and Y defined by p XY (x,y) the marginal distribution of X and Y are given by, p X XY(x,y) all y p (x) p Y XY(x,y) all x (y) p That is, just take row sums and column sums! Example Y 5% % 5% p X (x) X 5% % 5% p Y (y)

23 Example What are the marginals? X Y marginally, both X and Y have the Bernoulli distribution with p=.5: X~Bernoulli(.5), Y~Bernoulli(.5) 2.5 Conditional Probabilities Let D denote whether a HARP eligible home defaults ( is default and is no default). Let H denote whether a HARP eligible home took advantage of HARP ( is yes and is no) Here is the joint probability table: H D

24 H D It s a little harder to read the relationship directly off the numbers here. We need a way of describing the dependence. Conditional probabilities do this by answering the question: If you know the outcome for one variable (say whether or not the home took advantage of HARP), how does that change the probabilities associated with the other variable? Consider the question If the home took advantage of HARP, what is the chance that the home defaults? To see the answer to this question suppose there are, homes. In this case, we know the number of homes that fall in each of the four categories (we just multiply the probability times the total number of homes). H D

25 How many homes took advantage of HARP Ans: 2 Out of the homes that took advantage, what fraction defaulted? Ans: 8/2=.5% H D H D So, out of the 2 homes that took advantage, only 8 defaulted or.5% This is the same as asking out.2 fraction of the homes that took advantage of HARP, what fraction of those homes defaulted? Ans:.8/.2=.5% 25

26 In general, for random variables X and Y,we ask, what is Pr(Y=y) given we know X=x. Out of the times X=x, what fraction also have Y=y? Mathematically this looks like: p XY(x,y) p (x) X Joint probability = how often we get X=x and Y=y how often we get X=x Marginal probability Conditional probability and Conditional Distributions We just answered the question what is the probability a home defaults, given that it took advantage of HARP. We could also calculate the probability that house doesn t default given that it took advantage of HARP. This will give us the model for defaults given the home took advantage of HARP. We call this a conditional distribution or conditional model for default given the home took advantage of HARP. 26

27 This conditional model is denoted using a vertical bar Notation YX x is sometimes used as a symbol for the conditional distribution of Y give X=x. The conditional distribution is the model we use for Y given X=x This is different but could be compared to the unconditional distribution of Y, that is, the model we use for Y when we are not told what happened for x. H D D Pr(D H=) Marginal or unconditional distribution of default Conditional distribution of given home takes HARP Interpretation: If a home took HARP, there is a.5% chance the home defaults. Of all HARP eligible homes, there is a 2.98% chance of default. Notice that the conditional distribution is a real distribution that sums to one. It is the distribution of default given the home took HARP. 27

28 Example X 5% % 5% p Y (y) Y 5% % 5% p X (x) What is Y X=.5? What is Y X=.5? y.5..5 p(y.5) y.5..5 p(y.5) Independence In our HARP example, did knowing that a home took advantage of HARP change the default probability? YES This makes us think they have something to do with each other. Learning X=x, changed our probabilities for Y. There was information in X=x about Y. 28

29 In general, given two random variables, our beliefs about the probability of Y may or may not change when we learn of the outcome for X! Consider the joint distribution of whether stocks go up or down today S i and whether stocks go up or down tomorrow S i+ : S i S i What is the dist of S i+ given S i =? S i+ p(s i+ ).25/.5 =.5.25/.5 =.5 What is the dist of S i+ given S i =? S i+ p(s i+ ).25/.5 =.5.25/.5 =.5 S i+ S i What is the marginal p(s i+ )? S i+ p(s i+ ) = =.5 29

30 Wow! All three of p(s i+ ), p(s i+ ), and p(s i+ ) are the same! What does this mean? The probability that stocks go up tomorrow is not changed when we find out if stocks went up or down today. There is no information in up or down stock price moves today about whether stocks go up or down tomorrow. They have nothing to do with each other! When two thing have nothing to do with each other like this we say they are independent. 3

31 Independence Let X and Y be discrete random variables. If pyx ( y x) py( y) for all x,y we say the random variables are independent. An equivalent definition of Independence Suppose X and Y are independent. Then, p ( y) p ( y x) Y Y X pxy(, x y) p ( x) X so, p (, x y) p () y p () x XY Y X The joint is the product of the marginals. Example the two coins again: X Y

32 Example X 5% % 5% p Y (y) Clearly, X and Y Y are not independent in this example! 5% % 5% p X (x) What is Y X=.5? What is Y X=.5? y.5..5 p(y.5) y.5..5 p(y.5) A useful modeling tricks. We have: p YX ( y x) pxy ( x, y) p ( x) There are three probabilities in the above relationship: pyx ( y x ), px ( x ), and pxy ( x, y) Given any two we can recover the third. For example, if we have a model for the marginal and a model for the conditional, we can recover the joint as p ( x, y) p ( y x) p ( x) XY Y X X X 32

33 A similar result holds for more than two variables: p( X, X, X ) p X p X X p X X, X This is a nice way to model time series data.,x,..., X X,...,..., P X P X P X P X X X P X X X X 2 T T 2 T P Xt X, X2..., Xt is the distribution of X t given the past. If I know this, knowledge of past values of x tells us what the distribution of the next value X is. So this says that we should be able to figure out the probability of the sequence X, X 2,,X T if we just knew all the conditionals on the right hand side. This doesn t seem easier, but with some additional assumptions it can be! Lets consider a couple of special cases of the dependence structure. The idd model The Markovian model. 33

34 2.8 A special case: The IID Model Let s consider the case of three variables, X, X 2, and X 3 that denote the outcomes of three consecutive coin tosses. The marginal (or unconditional) distribution for each coin toss is a Bernoulli(.5) We also know the coin tosses are independent so that the model for any one coin toss is not dependent on what happened on the other coin tosses. P(X 3 X 2,X )=P(X 3 ) So each outcome has the same probability model (always a.5 chance of a head and.5 chance of a tail) Each outcome is independent (these are coin tosses!) from one and other. 34

35 Consider the case of 2 coin tosses To model how we think about the coins we have: X ~ Bernoull(.5) X 2 ~ Bernoull(.5) and X is independent of X 2 We say the two X s are independent and identically distributed, X i ~Bernoull(.5) They are iid independent (the first i) identically distributed (the id) 35

36 To model the tossing of n coins: Let X i denote the outcome of the i th coin, i=,2,...n We say the X i are iid. The outcome for each coin is independent of the outcome for all the others. For each coin X i ~ Bernoulli(.5) Suppose you get heads in a row. What is the prob the next one is a head? 36

37 iid Random Variables In general if we say X, X, 2 X n are iid, we mean that each is independent of all the others, and they each have the same probability distribution. Notice that the X i can follow Special about the Bernoulli! model, there is nothing Remember, the rv s refer to the possible outcomes before they happen. The rv s describe what can happen and the probability tells us how likely each outcome is. Alternatively, we can observe data or the outcome or realization of a r.v. Sometimes we refer to iid data as draws from an iid r.v. 37

38 Example Suppose we consider whether or not mortgages default. X i X i if the i th mortgage defaults if the i th mortgage doesn t default To say the X s are iid Bernoull(.) Says a lot. It can be a way to summarize what you have seen: the numbers we have already seen look like draws from the common distribution Default and it can tell us what we expect to see in the future (if things don t change of course). We are using the idea of iid rv s as a model for something in the real world. 38

39 Example How do you think about tosses of a die? Let Y i denote the outcome for the i th toss. The Y s are iid with, y p(y) for each die draws could look like this: tosses Index Example Stock price moves Series equals if up and if down

40 C5Does this data Example. look iid?.5. Index Does this data look iid? Intuitively the iid model is meant to describe numbers that (i ) have no pattern, are random (_id) but, over the long haul, the probs of the distribution tell you how often certain values (or sets of values) occur. There is no pattern to coin tosses, but over the long haul you get about half heads. 4

41 What is the probability of a given sequence of outcomes for the iid model? Let s go back to our modeling approach of last section and ask what happened to the following expression if the X s are iid. P X, X2,..., XN P X P X2 X P X3 X, X2... P XN X, X2..., XN Since the X s are independent, the conditional probabilities are just the unconditional: etc P X2 X P X2 P X3 X, X2 P X3 So P X, X,..., X P X P X... P X 2 N 2 N OK, so the X s are iid and therefore they are independent. In this case: P X, X2,..., XN P X P X2... P XN Since they are iid they all have the same probability models. That is, the model for X is the same as the model for X 2 and so on. They are all Bernoulli(.5). 4

42 What is the chance of getting a head followed by 2 tails on three tosses of a coin? P X, X, X P X X X.5*.5* X i ~ Bernoull(.5) What is the chance of getting 2 heads followed by 2 tails? If mortgage defaults are iid Bernoulli(.2), what is the chance of the first 5 not defaulting and the 6 th mortgage defaulting?,,,,, P X X X X X X

43 A special case (non iid): the Markovian Model A simple time series model might say that Y t depends only on the most recent Y t, but not any others. In this case we have: PYY, 2,..., YN PY PY 2 Y PY 3 YY, 2... PY N YY, 2..., YN becomes PYY, 2,..., YN PY PY 2 Y PY 3 Y2... PY N YN If it is the case that P Yi Yi is the same for all i then we only need to specify a model for P Yi Yi in order go get PYY, 2,..., YN Example: Let Y i be an indicator for whether the i th trade is buyer or seller initiated. Y i = denotes buyer and Y i = denotes seller initiated. We might be able to model it as Markovian. Will there be persistence in the process, i.e. will a buy tell you something different about the next trade than a sell? 43

44 y P(y) y P(y i y i =) y P(y i y i =) Then, for example, P Y, Y2, Y3, Y4.5*(2/3)*(/3)*(2/3) We could figure out the probability of any sequence of outcomes! 2.9 Models and Formulas We often use mathematical formulas to describe how numerical quantities are related. We can do this with our models as well. Example Suppose you are playing a game where you toss a coin and win $ if it comes up heads and lose $ if it comes up tails. Let W denote your winnings. 44

45 What is the distribution of W? w: p(w):.5.5 We can represent this in another way using the Bernoulli distribution. Let X~Bernoulli(.5). W = + 2 X Whatever X turns out to be, the formula gives the W. Example Let R be your uncertain return. Suppose you invest $ thousand. How is your end of period wealth related to R? Example Suppose you toss two coins: X and X 2 ~ Bernoulli(.5) iid. Let Y = X + X 2. What does Y mean? 45

46 2. The Binomial Distribution We have seen how these probability models can be used to think about coin tosses, die tosses, and defects. We use probability to model a wide variety of phenomena in the real world. There are many type of distributions that are useful for various situations. Our most basic type of distribution is the Bernoulli. In the section we learn about the Binomial. In the next section, we will consider additional models. Let X, X 2,,X n denote n iid Bernoulli(p) random variables Let Y X X X X 2 What does Y mean? n n i i) You try something n times (X i denotes the i th outcome) ii) Each time you have the same chance p of success (X i is one for a success and zero for a failure) iii) Each time you try your probability of a success does not depend on any of the other outcomes. iv) Y simply counts the number of successes in n tries. i 46

47 Binomial Distribution The binomial distribution is the probability distribution for the total number of successes: Y X X X X 2 Example: What is the probability of succeeding 4 times in 5 (independent) tries when the probability of a success on any given try is.3 (p=.3, n=5)? The Binomial distribution answers this question. n n i i More examples How many heads do I get when I toss a coin times? If Kobe Bryant makes 82 percent of his free throw shots how many does he make in attempts (assuming the outcomes are independent!). 47

48 Suppose n=2: (x,x 2 ) p(x,x 2 ) y (,).25 (,).25 (,).25 (,).25 2 X 2 X Suppose X, X 2, X n are iid Bernoulli(p). Then Y X X2 X n Has the Binomial distribution with parameters n and p. We write: Y ~ B(n, p) X i tells you whether it happened on the i th trial. Y is the total number of times it happened out of n trials. 48

49 There is a formula giving the binomial probabilities: n! y n y py ( y) p ( p) y,,2, n ( n y)! y! Probability of getting number of ways to y successes on n tries get y successes on (one way only) n tries. where n! = n(n )(n 2)(n 3)...(3)(2)(). Example B(,.2) B(,.5) B(,.8).3.2 pp2.. 5 y 49

50 Example A firm was being sued for sexual discrimination. As a (small) part of the evidence the following data was used. Each point corresponds to a firm in the same industry. The x axis give the number of partners. The y axis gives the number of female partners. yy nn 9 This point corresponds to the firm in question. Clearly the point corresponding to the firm looks unusual. How can we quantify this? If whether a partner is male or female is iid Bernoulli(p) then the total number of female partners at the i th firm should be B(n i,p) where n i is the number of partners at the i th firm. 5

51 What should we use for p? Not counting the firm in question, 7% of partners are female. Let s estimate p=.7. (But, we could be wrong!!) y p(y) pyf p(y) for Y~B(85,.7) yf Under our assumptions, the prob of having female partners at the firm with 85 partners is

52 2. Another non iid Model, the Random Walk At left is a plot of the price of a stock. The price is recorded every time it changes. Each price change is one tick which in this case is Does this data look iid?? The trick here is to look at the price changes: D P P t=2,3,4,... t t t The D t look i.i.d. with Pr(D t =.)=.55 Pr(D t =.)=.45 52

53 What is p(p p,p, p ) t t t? Our model is, P P D with D t+ : t t t d p(d) and the D's are iid. What is the conditional probability distribution of P t+ P t =p t? P t+ : p t+ p(p t+ p t ) p t..45 p t Given our model, how would you predict the next price? The last price in the series is.2. 53

54 Data that kind of wanders can often be modeled as a random walk. P P D t t t where the D s look iid from some distribution. The next value is the current value plus a random increment. 54