Statistics and Econometrics I
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1 Statistics and Econometrics I Random Variables Shiu-Sheng Chen Department of Economics National Taiwan University October 5, 2016 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
2 Part I Random Variables Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
3 Random Variables Random Variables: A Loose Definition Definition A random variable is a variable that takes values according to a certain probability distribution. Keys to know: All the possible values it can take. The probability distribution according to which it takes all possible values. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
4 Random Variables Random Variables: A Formal Definition Definition A random variable X is a real-value function from the state space to the real numbers: X : Ω R and it assigns one and only one real number to each element ω Ω. The set of all possible values that a random variable X can take is called the range of X. We use small letter x to denote the possible value of a random variable X: X(ω) = x Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
5 Random Variables Random Variables According to the range of a random variable, we can define two types of random variables. Discrete random variables: The range is finite or countably infinite. 1 (1) The number of defective light bulbs in a box of ten (finite) (2) The number of tails until the first heads comes up (countably infinite) Continuous random variables: The range is uncountable (any value in an interval) (1) Consider the experiment where a light bulb is tested until failure, and let X denote the time to failure. Range of X = [0, ) 1 A set of elements is countably infinite if the elements in the set can be put into one-to-one correspondence with the positive integers. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
6 Part II Discrete Random Variables Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
7 Discrete Random Variables Discrete Random Variables How to Assign Probability? Let X be a discrete random variable, the probability of X = x is P (X = x) = P ({ω : X(ω) = x}) Probability of X = x is indeed the probability that the event ω occurs, where ω is the event such that X(ω) = x. That is, the former probability is induced by the latter. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
8 Discrete Random Variables Example: Flip a Fair Coin Twice The sample space is Ω = {HH, HT, T H, T T } Let X be the number of heads. The range of X is {0, 1, 2} For example, P (X = 1) = P ({ω : X(ω) = 1}) = P ({HT }) + P ({T H}) = = 1 2 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
9 Discrete Random Variables Probability Distribution Although it is important to understand the relationship between (1) the probability function defined on the original sample space, and (2) the (induced) probability for a random variable, we will no longer refer to the original probability structure when considering a random variable. Instead, we simply describe the probability distribution of a random variable. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
10 Discrete Random Variables Probability Distribution Definition (Probability Distribution) Let X be a random variable. The probability distribution of X is to specify all probabilities involving X. There are two important ways to specify the probability distribution of a discrete random variable (1) Probability Mass Function (2) Cumulative Distribution Function Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
11 Discrete Random Variables Probability Mass Function Definition (pmf) Given a discrete random variable X. A probability mass function (pmf), f(x) : R [0, 1] is defined by f(x) = P (X = x) A probability mass function is also called a discrete probability density function (discrete pdf). The possible realizations, x, are called mass points of X. A preferable notation: f X (x) Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
12 Discrete Random Variables Probability Mass Function Define the support of X: supp(x) = {x : f(x) > 0} Properties: x supp(x) f(x) = 1 P (X A) = x A f(x), A supp(x) Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
13 Discrete Random Variables Example: Flip a Fair Coin Twice The sample space is Ω = {HH, HT, T H, T T } Let X be the number of heads. Table: The Original Probability Structure vs. Probability Distribution of X ω P ({ω}) X(ω) T T 1/4 0 T H 1/4 1 HT 1/4 1 HH 1/4 2 x f(x) = P (X = x) 0 1/4 1 1/2 2 1/4 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
14 Discrete Random Variables Example: Flip a Fair Coin Twice The pmf can be rewritten as 1 4, x = f(x) =, x = 1 1 4, x = 2 0, otherwise The support is supp(x) = {0, 1, 2}, f(x) = f(0) + f(1) + f(2) = 1/4 + 1/2 + 1/4 = 1 x supp(x) Let A = {X 1} = {0, 1} P (X A) = x A f(x) = f(0) + f(1) = 1/4 + 1/2 = 3/4 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
15 Discrete Random Variables Probability Mass Function Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
16 Discrete Random Variables Cumulative Distribution Function An alternative way to specify the probability distribution is to give the probabilities of all events of the form {X x}, x R For example, what is the probability that the resulting number by rolling a die is smaller than 3.8? This leads to the following definition of cumulative distribution function. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
17 Discrete Random Variables Cumulative Distribution Function Definition (CDF) Given any real variable x, a function F (x) : R [0, 1]: F (x) = P (X x) is called a cumulated distribution function (CDF), or distribution function. A preferable notation: F X (x) Clearly, F (x) = P (X x) = w x P (X = w) Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
18 Discrete Random Variables Example: Flip a Fair Coin Twice Given the pmf of X 1 4, x = f(x) =, x = 1 1 4, x = 2 0, otherwise What is F ( 2)? What is F (x) = 0 for all x < 0? What is F (1.5)? Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
19 Discrete Random Variables Example: Flip a Fair Coin Twice Hence, the CDF is 0, x < F (x) = P (X x) =, 0 x < 1 3 4, 1 x < 2 1, 2 x Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
20 Discrete Random Variables Cumulative Distribution Function Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
21 Discrete Random Variables Cumulative Distribution Function Theorem (Properties of CDF) Let F (x) be the CDF of a random variable X. Then, If a < b, then F (a) F (b) and P (a < X b) = F (b) F (a). lim x F (x) = 0, and lim x F (x) = 1 F (x) = lim δ 0 F (x + δ) Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
22 Part III Continuous Random Variables Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
23 Continuous Random Variables A random variable is called continuous if it can take all possible and infinitely many values in the range of the random variable. The sample space is no longer countable. For instance, The computer time (in seconds) required to process a certain program Stock returns The percentage of exam complete after 1 hour Technically, without limitations caused by rounding to a certain number of digits, we could imagine that any real number could provide a feasible outcome for above examples. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
24 Continuous Random Variables Though a continuous variable can take any possible value in an interval, its measured value cannot. This is because no measuring device has infinite precision. Thus, continuous variables do not really exist in real life; they are only ideal versions of the discretized variables which are measured. Nevertheless, the study of continuous random variables is meaningful as it provides useful, and quite accurate, approximations to probabilities pertaining to their discretized versions. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
25 How to Assign Probability? Discrete: flip a coin or roll a die. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
26 How to Assign Probability? Discrete: flip a coin or roll a die. How about spinning a spinner? Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
27 How to Assign Probability? Discrete: flip a coin or roll a die. How about spinning a spinner? Let X be the result of the spin. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
28 How to Assign Probability? Discrete: flip a coin or roll a die. How about spinning a spinner? Let X be the result of the spin. Warning! Impossible to assign each outcome positive probability. Why? Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
29 How to Assign Probability? Discrete: flip a coin or roll a die. How about spinning a spinner? Let X be the result of the spin. Warning! Impossible to assign each outcome positive probability. Why? Well, suppose NOT, and let the spinner be fair. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
30 How to Assign Probability? Discrete: flip a coin or roll a die. How about spinning a spinner? Let X be the result of the spin. Warning! Impossible to assign each outcome positive probability. Why? Well, suppose NOT, and let the spinner be fair. Each outcome has probability p > 0. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
31 How to Assign Probability? Discrete: flip a coin or roll a die. How about spinning a spinner? Let X be the result of the spin. Warning! Impossible to assign each outcome positive probability. Why? Well, suppose NOT, and let the spinner be fair. Each outcome has probability p > 0. Let A Ω be an event that contains n distinct outcomes. Choose n large enough s.t. p > 1 n. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
32 How to Assign Probability? Discrete: flip a coin or roll a die. How about spinning a spinner? Let X be the result of the spin. Warning! Impossible to assign each outcome positive probability. Why? Well, suppose NOT, and let the spinner be fair. Each outcome has probability p > 0. Let A Ω be an event that contains n distinct outcomes. Choose n large enough s.t. p > 1 n. Then P (X A) = np > 1 Big Trouble! Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
33 How to Assign Probability? Hence p must be zero! That is, if X is a continuous random variable, P (X = a) = 0 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
34 How to Assign Probability? Hence p must be zero! That is, if X is a continuous random variable, P (X = a) = 0 How can P (X = a) = 0 make sense? Can many nothings make something? Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
35 How to Assign Probability? Hence p must be zero! That is, if X is a continuous random variable, P (X = a) = 0 How can P (X = a) = 0 make sense? Can many nothings make something? Think about the length of a point vs. the length of an interval. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
36 How to Assign Probability? Hence p must be zero! That is, if X is a continuous random variable, P (X = a) = 0 How can P (X = a) = 0 make sense? Can many nothings make something? Think about the length of a point vs. the length of an interval. A Zero-probability event is NOT an impossible event. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
37 Continuous Random Variables Now we formally define a continuous random variable via its distribution function. In general, if the support for a continuous random variable is not specified, we assume that supp(x) = {x : < x < } Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
38 Continuous Random Variables Definition (Continuous RV and Probability Density Function) Let F (x) = P (X x) be the distribution function of a random variable X. Then X is continuous if F (x) is a continuous function of x There is a function f( ) such that F (x) = x f(u)du We call f( ) the probability density function. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
39 The Distribution Function F ( ) = 0, F ( ) = 1, F ( ) monotonic nondecrease 1 F(x) 0 x Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
40 The Distribution Function F (x) = x f(u)du Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
41 Probability Density Function The probability density function (pdf) has following properties: (a) Marginal rate of growth (b) Nonnegative f(x) = df (x) dx f(x) 0, x (c) The integral over the support of X is one f(x)dx = F ( ) = 1 (d) For every interval (bounded or unbounded), A, the probability that X takes a value in the interval is the integral of f(x) over the interval. P (X A) = f(x)dx Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36 A
42 Probability Density Function For example, A = [a, b], P (X A) = P (a X b) = b a f(x)dx Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
43 Probability Density Function For example, A = [a, b], P (X A) = P (a X b) = b a f(x)dx Since P (X = c) = 0 for any real value c, P (a X b) = P (a < X < b) = P (a X < b) = P (a < X b) }{{} =F (b) F (a) Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
44 Example Suppose that a random variable X has pdf f(x) = 1 with supp(x) = {x 0 x 1}. Clearly, 1 0 f(x) 0 f(x)dx = 1 Moreover, 0 x < 0 F (x) = x 0 x 1 1 x > 1 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
45 Quantiles Definition (Quantiles) Let X be a continuous random variable with CDF F (x). Suppose that F (π p ) = P (X π p ) = p, and F is strictly increasing, π p = F 1 (p) is called the p-th quantile of X. F 1 ( ) is called the inverse CDF or quantile function. Given p = 0.5, the 2nd quantile, π 0.5, is called the median. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
46 Probability Mass Function vs. Probability Density Function pmf (discrete pdf): f(x) : R [0, 1], f(x) = P (X = x) pdf: f(x) : R R +, f(x) P (X = x) That is, density is not probability. Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
47 Notations Random variable X has pmf (pdf), f(x) CDF, F (x) mean, E(X) = µ variance, V ar(x) Notations X f(x) X F (x) X (µ, σ 2 ) Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, / 36
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