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8 - Lecture 3 Part B

# 8 - Lecture 3 Part B - Introduction to Econometrics Econ...

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Introduction to Econometrics Econ 322; Summer 2011 Ruby HENRY [email protected] June 8, 2011 Ruby HENRY () Introduction to Econometrics June 8, 2011 1 / 1

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The χ 2 distribution The Normal distribution is an important distribution for many reasons. However, the Normal distribution is practical only for random variables that can take both positive and negative values. There are many occasions in this course that we will deal with random variables that can only take positive values. The following two probability distributions are de°ned only for positive values: Ruby HENRY () Introduction to Econometrics June 8, 2011 2 / 1
The χ 2 distribution The Normal distribution is an important distribution for many reasons. However, the Normal distribution is practical only for random variables that can take both positive and negative values. There are many occasions in this course that we will deal with random variables that can only take positive values. The following two probability distributions are de°ned only for positive values: The random variable Y has a Chi-Squared distribution with m degrees of freedom ( Y ° χ 2 ( m ) ) if Y is the sum of m standard normal random variables squared. That is Y ° χ 2 ( m ) if Y = X 2 1 + . . . + X 2 m where X 1 , . . . , X m are all N(0,1). Ruby HENRY () Introduction to Econometrics June 8, 2011 2 / 1

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The F distribution A closely related distribution is the ±F² distribution. Ruby HENRY () Introduction to Econometrics June 8, 2011 3 / 1
The F distribution A closely related distribution is the ±F² distribution. In this course we de°ne the F m , which is de°ned as Y ° F m , if Y = X m where X ° χ 2 ( m ) . Ruby HENRY () Introduction to Econometrics June 8, 2011 3 / 1

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The F distribution A closely related distribution is the ±F² distribution. In this course we de°ne the F m , which is de°ned as Y ° F m , if Y = X m where X ° χ 2 ( m ) . That is, an F random variable is a Chi-squared random variable divided by its degrees of freedom. Ruby HENRY () Introduction to Econometrics June 8, 2011 3 / 1
The Student-t Distribution Another distribution we will come across is the The Student-t Distribution Ruby HENRY () Introduction to Econometrics June 8, 2011 4 / 1

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The Student-t Distribution Another distribution we will come across is the The Student-t Distribution Let Z ° N ( 0 , 1 ) and let Y ° χ 2 m be two independent random variables. The t = Z p Y / m ° t m . Ruby HENRY () Introduction to Econometrics June 8, 2011 4 / 1
The Student-t Distribution Another distribution we will come across is the The Student-t Distribution Let Z ° N ( 0 , 1 ) and let Y ° χ 2 m be two independent random variables. The t = Z p Y / m ° t m . This distribution looks a lot like the standard normal but it has ±fatter² tails. As m ±! the t-distribution approaches the standard Normal. Ruby HENRY () Introduction to Econometrics June 8, 2011 4 / 1

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Random Sampling and the Sample Mean We typically do not get to observe the whole population of a random variable. Therefore we have to make do with a sample from the population.
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