Review of Basic Statistics , Lecture notes of Social Statistics and Data Analysis

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Review of Probability and

Statistics

(i.e. things you learned in Ec 10 and need to remember to do well in this class!)

Random Variables

X is a random variable if it represents a random draw from some population a discrete random variable can take on only selected values a continuous random variable can take on any value in a real interval associated with each random variable is a probability distribution

Expected Value of X – E(X)

The expected value is really just a probability weighted average of X E(X) is the mean of the distribution of X, denoted by x Let f(xi) be the probability that X=xi, then     n i X i i E X x f x 1  ( ) ( )

Variance of X – Var(X)

The variance of X is a measure of the dispersion of the distribution Var(X) is the expected value of the squared deviations from the mean, so

2 2

X X

  Var X  E X  

Covariance – Cov(X,Y)

Covariance between X and Y is a measure of the association between two random variables, X & Y If positive, then both move up or down together If negative, then if X is high, Y is low, vice versa     XY X Y

  Cov ( X , Y )  E X   Y  

Correlation Between X and Y

Covariance is dependent upon the units of X & Y [Cov(aX,bY)=abCov(X,Y)] Correlation, Corr(X,Y), scales covariance by the standard deviations of X & Y so that it lies between 1 & –   2 1 ( ) ( ) ( , ) Var X Var Y Cov X Y X Y XY XY      

Properties of Expectations

E(a)=a, Var(a)= E(X)=X, i.e. E(E(X))=E(X) E(aX+b)=aE(X)+b E(X+Y)=E(X)+E(Y) E(X-Y)=E(X)-E(Y) E(X- X)=0 or E(X-E(X))= E((aX) 2 )=a 2 E(X 2 )

More Properties

Var(X) = E(X 2 ) – x 2 Var(aX+b) = a^2 Var(X) Var(X+Y) = Var(X) +Var(Y) +2Cov(X,Y) Var(X-Y) = Var(X) +Var(Y) - 2Cov(X,Y) Cov(X,Y) = E(XY)-xy If (and only if) X,Y independent, then  Var(X+Y)=Var(X)+Var(Y), E(XY)=E(X)E(Y)

The Standard Normal

Any random variable can be “standardized” by subtracting the mean, , and dividing by the standard deviation,  , so E(Z)=0, Var(Z)= Thus, the standard normal, N(0,1), has pdf   2 2 2 1 z z e    

Properties of the Normal

If X~N(,^2 ), then aX+b ~N(a+b,a^2 ^2 ) A linear combination of independent, identically distributed (iid) normal random variables will also be normally distributed If Y 1 ,Y 2 , … Yn are iid and ~N(, 2 ), then         n ~ N , 2  Y 

The Chi-Square Distribution

Suppose that Zi , i=1,…,n are iid ~ N(0,1), and X=(Zi^2 ), then X has a chi-square distribution with n degrees of freedom (df), that is X~ 2 n If X~ 2 n, then E(X)=n and Var(X)=2n

The t distribution

If a random variable, T, has a t distribution with n degrees of freedom, then it is denoted as T~tn E(T)=0 (for n>1) and Var(T)=n/(n-2) (for n>2) T is a function of Z~N(0,1) and X~^2 n as follows:

n

X

Z

T 

Random Samples and Sampling

For a random variable Y, repeated draws from the same population can be labeled as Y 1 , Y 2 ,... , Yn If every combination of n sample points has an equal chance of being selected, this is a random sample A random sample is a set of independent, identically distributed (i.i.d) random variables

Estimators and Estimates

Typically, we can’t observe the full population, so we must make inferences base on estimates from a random sample An estimator is just a mathematical formula for estimating a population parameter from sample data An estimate is the actual number the formula produces from the sample data