1 Chapter 1 An Introduction to statistics AnIntroductionToStatistics tex or pdf Edward Morey August 30 2010 1 1 The name of this course is Mathematical Statistics for Economics Econ 7818 So this is a course in statistics What are statistics Statistics is the plural of the word statistic De nition 1 A statistic is a function of one or more random variables To know exactly what this means one must rst de ne a variable and then a random variable Put simply a variable is something that varys That is a variable can take on di erent numerical values a realized value of a variable is some number Each number represents a distinct state For example the variable G might represent gender where 1 corresponds to the state female and 0 corresponds to the state not female Or for example if X is a variable such that 0 x 123 X can take any numerical value between zero and 123 inclusive the variable X might represent age of a human Here I have assumed all humans are not younger than zero and not older than 123 years everyone would not agree on this lower bound 1 X is the name of the random variable e g age price or amount of sexual activity and x is a numerical value of X 2 1 From Wikipedia The longest unambiguously documented lifespan is that of Jeanne Calment of France 1875 1997 who died at age 122 years and 164 days She met Vincent van Gogh at age 14 1 This led to her being noticed by the media in 1985 at age 110 Subsequent investigation found that her life was documented in the records of her native city of Arles beyond reasonable question 2 More evidence for the Calment case has been produced than for any other supercentenarian case which makes her case a standard among the oldest people recordholders citation needed http en wikipedia org wiki Oldest people 2 The issue of how to distinguish between a random variable and a speci c realization of that random variable can be confusing and the literture is not consistent in how it notationally distinguishes between the two I won t be either I will try to use uppper case to denote the name of a random variable and lower case to denote a speci c value of that random variable However I and others might use x to refer to both and hope the reader can determine which is meant by the context 1 To say that the value of a rv can be expressed with a number is not vary restrictive if X has cardinal properties e g if X represents age the the numbers have cardinal meaning if X only has meaning in terms of a ranking e g class rank then only the ordinal properties of the numbers are important if X simply denotes categories e g gender or race then the numbers mean nothing other than di erent numbers represent di erent categories 3 Let s assume X is a random variable I will de ne random variable in a second In which case y f x g x and m 4 7x are each statistics all of the same random variable X The letters f and g are the names of particular functions Or more generally imagine three random variables X Y and Z In which case x y z b h x y z 1 1 x y z 2 2 x y z and are each statistics So c m b is a rv Alternatively one could let x refer to the rv and let xi refer to a value i of the rv This approach will be pretty clear if there is only one rv being considered But what if there are three rvs Do I give them di erent names like x y and z If so zi refers to a realized value of z But what if instead of x y and z I had denoted the three random variables in the text x1 x2 and x3 where the subscripts now refer to di erent rv s not di erent observation on x One must be viligant We need to be careful and gure out what is going on by the context Being explicit about what we mean is also a good thing if we don t want the reader to get confused 3 You are probably ready to conclude that I like footnotes I do they allow me to digress a former student accused me of being the King of digression and tangents I discuss the properties of numbers in Morey confuser surplus Latent Gold a statistical program I use allows one to change the speci cation of random variables between cardinal ordinal and nominial categorical The di erences between many statistical and economic models are often only the numerical properties of the dependent and independent variables 2 All statistics are random variables but all random variables are not statistics unless one de nes x x as a function 1 2 So what is a random variable De nition 2 X is a random variable if it is a variable and if it has a distribution Said another way X is a random variable if 8 a and b one can determine the probability that a x b if one knows the distribution of X Note that X takes speci c values e g if X is weight each of us has a speci c weight but weight in the population has some distribution The above de nition is not self contained It requires that we know what a distribution is and we have yet to de ne that term other than we have de ned it as something that allows us to calculate Pr a x b Also note the de ntion requires that X has a distrbution but it does not require that we know what distribution The book Introduction to the Theory of Statistics Mood Graybill and Boes de nes a continuous random variable as follows The variable X is a one dimensional continuous random variable if there exists a function f x such that f x 0 8 x in the interval 1 x 1 and the probability that a x b is4 Pr a x b Zb f x dx a The function f X is called a density function or a probability density function The function f X describes the distribution of X Any function f X can serve as a density function as long as f x and5 0 1 Z 1 x 1 f x dx 1 1 4 Note the quali ying adjective continuous that f x 1 is not a requirement necessary condition It is required for certain types of density functions but not all of them What types What is required is that 1 Rb R f x dx 1 which follow from the restriction that f x dx 1 5 Note a 1 3 Why do we care about density functions Economic models typically assume outcomes how much you drink whether the interest rate will rise are the result of …