Statistics Review, Study notes of Statistics

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Statistics Review

Christopher Taber

Wisconsin

Spring Semester, 2011

Outline

1 Random Variables

2 Distribution Functions

3 The Expectation of a Random Variable

4 Variance

5 Continuous Random Variables

6 Covariance and Correlation

7 Normal Random Variables

8 Conditional Expectations

Random Variables

Random variables

Lets forget about the details that arise in dealing with data for a while. Most objects that economists think about are random variables.

Informally, a random variable is a numerical outcome or measurement with some element of chance about it. That is, it makes sense to think of it as having possibly had some value other than what is observed.

Econometrics is a tool that allows us to learn about these random variables from the data at our disposal.

Random Variables

Examples of random variables:

Gross Domestic Product Stock Prices Wages of Workers Years of Schooling Attained by Students Numeric Grade in a Class Number of Job Offers Received Demand for a new product at a given price

Distribution Functions

Outline

1 Random Variables

2 Distribution Functions

3 The Expectation of a Random Variable

4 Variance

5 Continuous Random Variables

6 Covariance and Correlation

7 Normal Random Variables

8 Conditional Expectations

Distribution Functions

Probability Density Functions

Suppose that X is a random variable that takes on J possible values x 1 , x 2 , ...xJ.

The probability density function (pdf), f (·) of X is defined as:

f (xj) = Pr(X = xj)

Some conventions: capital letters are used to denote the variable, small letters realizations or possible values; a pdf is a lower-case letter (often f )

Now it follows that if X can only take on the values x 1 , x 2 , ...xJ , we have ∑J

j= 1

f (xj) = 1

The Expectation of a Random Variable

Outline

1 Random Variables

2 Distribution Functions

3 The Expectation of a Random Variable

4 Variance

5 Continuous Random Variables

6 Covariance and Correlation

7 Normal Random Variables

8 Conditional Expectations

The Expectation of a Random Variable

The expectation is also called the mean or the average.

E(X) =

∑^ N

j= 1

xjf (xj)

(The expectation is thought of as a ‘typical value’ or measure of central tendency, though it has shortcomings for each of these purposes.)

The Expectation of a Random Variable

An example: Grades

Using the distribution from the example above, the expected grade is:

E(G) = 0. 3 × 4 + 0. 4 × 3 + 0. 25 × 2 + 0. 04 × 1 + 0. 01 × 0 = 2. 94

The Expectation of a Random Variable

Interpretation of Expectation as Bet

One interpretation of an expectation is the value of a bet. I would break even if the expected value of the bet was zero.

The Expectation of a Random Variable

Example 2: Die Roll

Suppose you meet a guy on the street who charges you $3. to play a game. He rolls a die and gives you that amount in dollars, i.e. if he rolls a 1 you get $1.00, etc. Is this a good bet?

The expected payoff from the bet is

E (Y − 3. 5 ) =

The Expectation of a Random Variable

Example 3: Occupation

Suppose you are choosing between being a doctor or a lawyer. You may choose to go into the profession where the expected earnings are highest.

The Expectation of a Random Variable

Estimation of Expected Value

You should have learned that the way we estimate the expected value is to use the sample mean

That is suppose we want to estimate E(X) from a sample of data X 1 , X 2 , ..., XN

We estimate using

X^ ¯ = 1 N

∑^ N

i= 1

Xi

The Expectation of a Random Variable

Example

Suppose data is (as in Wooldridge Example C.1) City Unemployment Rate 1 5. 2 6. 3 9. 4 4. 5 7. 6 8. 7 2. 8 3. 9 5. 10 7.