BS2247 Introduction to Econometrics Lecture 2: Fundamentals of - PowerPoint PPT Presentation

BS2247 Introduction to Econometrics Lecture 2: Fundamentals of Probability Dr. Kai Sun Aston Business School 1 / 30

Why do we care about this topic? ◮ Economic variables (e.g., education, wage, etc.) are random variables, in the sense that each observation (i.e., realization) is a random draw from the entire population. ◮ Each random variable has a probability measure. Roughly speaking, a probability measure is a function which maps from the occurring of an event (e.g., realization of a random variable) to the probability of the occurring of the event. 2 / 30

Why do we care about this topic? ◮ Economic variables (e.g., education, wage, etc.) are random variables, in the sense that each observation (i.e., realization) is a random draw from the entire population. ◮ Each random variable has a probability measure. Roughly speaking, a probability measure is a function which maps from the occurring of an event (e.g., realization of a random variable) to the probability of the occurring of the event. 2 / 30

For example, the probability measure can tell us that: The probability of, say, educ = 12 years (i.e., realization of a random variable, education) is, say, 0.2. This is the same as saying that 20% of the observations in the sample have educ = 12 years. 3 / 30

Discrete random variables ◮ Random variables that take on only a finite number of values ◮ For example, consider tossing a single coin. The two outcomes/events are heads and tails. ◮ A discrete random variable then can be defined as: x = 1 if the coin turns up heads, x = 0 if the coin turns up tails. By tossing the coin a number of times, we can calculate the probability of x = 1 and x = 0. 4 / 30

Discrete random variables ◮ Random variables that take on only a finite number of values ◮ For example, consider tossing a single coin. The two outcomes/events are heads and tails. ◮ A discrete random variable then can be defined as: x = 1 if the coin turns up heads, x = 0 if the coin turns up tails. By tossing the coin a number of times, we can calculate the probability of x = 1 and x = 0. 4 / 30

Discrete random variables ◮ Random variables that take on only a finite number of values ◮ For example, consider tossing a single coin. The two outcomes/events are heads and tails. ◮ A discrete random variable then can be defined as: x = 1 if the coin turns up heads, x = 0 if the coin turns up tails. By tossing the coin a number of times, we can calculate the probability of x = 1 and x = 0. 4 / 30

Probability Density Function (pdf) ◮ Generally, if a discrete random variable, X , takes on the n possible values { x 1 , . . . , x n } , then the probability measure is p i = P ( X = x i ), i = 1 , 2 , . . . , n , where 0 ≤ p i ≤ 1 and � i p i = 1. ◮ P ( · ) is also called probability density function (pdf) of X . “The probability of X = x i is equal to p i ”. 5 / 30

Probability Density Function (pdf) ◮ Generally, if a discrete random variable, X , takes on the n possible values { x 1 , . . . , x n } , then the probability measure is p i = P ( X = x i ), i = 1 , 2 , . . . , n , where 0 ≤ p i ≤ 1 and � i p i = 1. ◮ P ( · ) is also called probability density function (pdf) of X . “The probability of X = x i is equal to p i ”. 5 / 30

The pdf of heads and tails from the coin-tossing example This is essentially a histogram! 6 / 30

Continuous random variables ◮ They are random variables that take on numerous values. ◮ For example, wage should be a continuous random variable. ◮ In practice, education can also be considered as continuous. However, in theory, education may be discrete. 7 / 30

Continuous random variables ◮ The pdf of continuous random variable computes the probability of events (i.e., realizations) involving a range of values (not a particular value!). ◮ P ( a ≤ X ≤ b ) measures the probability that X ranges from a to b . ◮ For example, P (16 ≤ wage ≤ 18) = 0 . 1 says that the probability that wage ranges from 16 to 18 is 0.1. This is the same as saying that 10% of the observations in the sample have 16 / hour ≤ wage ≤ 18 / hour . 8 / 30

Continuous random variables ◮ The pdf of continuous random variable computes the probability of events (i.e., realizations) involving a range of values (not a particular value!). ◮ P ( a ≤ X ≤ b ) measures the probability that X ranges from a to b . ◮ For example, P (16 ≤ wage ≤ 18) = 0 . 1 says that the probability that wage ranges from 16 to 18 is 0.1. This is the same as saying that 10% of the observations in the sample have 16 / hour ≤ wage ≤ 18 / hour . 8 / 30

Continuous random variables ◮ The pdf of continuous random variable computes the probability of events (i.e., realizations) involving a range of values (not a particular value!). ◮ P ( a ≤ X ≤ b ) measures the probability that X ranges from a to b . ◮ For example, P (16 ≤ wage ≤ 18) = 0 . 1 says that the probability that wage ranges from 16 to 18 is 0.1. This is the same as saying that 10% of the observations in the sample have 16 / hour ≤ wage ≤ 18 / hour . 8 / 30

Use Histogram to illustrate the pdf of wage Histogram of wage 0.10 0.08 0.06 pdf of wage 0.04 0.02 0.00 0 10 20 30 wage 9 / 30

Use Density plot to illustrate the pdf of wage Density of wage 0.10 0.08 0.06 pdf of wage 0.04 0.02 0.00 0 10 20 30 wage 10 / 30

Normal Distribution Density of wage 0.10 0.08 0.06 pdf of wage 0.04 0.02 0.00 0 10 20 30 wage 11 / 30

Other Distributions 12 / 30

Cumulative Distribution Function (cdf) ◮ Sometimes it’s easier to work with cumulative distribution function (cdf) , defined as F ( x ) = P ( X ≤ x ) where X is a random variable (either discrete or continuous), and x is any real number. ◮ For continuous random variable, F ( x ) is the area under the pdf, to the left of the point x . 13 / 30

Cumulative Distribution Function (cdf) ◮ Sometimes it’s easier to work with cumulative distribution function (cdf) , defined as F ( x ) = P ( X ≤ x ) where X is a random variable (either discrete or continuous), and x is any real number. ◮ For continuous random variable, F ( x ) is the area under the pdf, to the left of the point x . 13 / 30

Cumulative Distribution Function (cdf) Properties of cdf: (1) P ( X > c ) = 1 − F ( c ), for any number c (2) P ( a < X ≤ b ) = F ( b ) − F ( a ) (3) for continuous random variable, any of the above inequality can become strict inequality, and vice versa From the previous example, if P (16 ≤ wage ≤ 18) = 0 . 1, then F (18) − F (16) = 0 . 1, where F is the cdf of wage. 14 / 30

Cumulative Distribution Function (cdf) Properties of cdf: (1) P ( X > c ) = 1 − F ( c ), for any number c (2) P ( a < X ≤ b ) = F ( b ) − F ( a ) (3) for continuous random variable, any of the above inequality can become strict inequality, and vice versa From the previous example, if P (16 ≤ wage ≤ 18) = 0 . 1, then F (18) − F (16) = 0 . 1, where F is the cdf of wage. 14 / 30

Cumulative Distribution Function (cdf) Properties of cdf: (1) P ( X > c ) = 1 − F ( c ), for any number c (2) P ( a < X ≤ b ) = F ( b ) − F ( a ) (3) for continuous random variable, any of the above inequality can become strict inequality, and vice versa From the previous example, if P (16 ≤ wage ≤ 18) = 0 . 1, then F (18) − F (16) = 0 . 1, where F is the cdf of wage. 14 / 30

Cumulative Distribution Function (cdf) Properties of cdf: (1) P ( X > c ) = 1 − F ( c ), for any number c (2) P ( a < X ≤ b ) = F ( b ) − F ( a ) (3) for continuous random variable, any of the above inequality can become strict inequality, and vice versa From the previous example, if P (16 ≤ wage ≤ 18) = 0 . 1, then F (18) − F (16) = 0 . 1, where F is the cdf of wage. 14 / 30

Features of Random Variables Expected value (population mean) ◮ It is a weighted average of all possible values of X . The weights are determined by pdf. ◮ Precisely, if X is discrete and can take values { x 1 , . . . , x n } with pdf p i = P ( X = x i ), then the expected value of X is E ( X ) = x 1 p 1 + · · · + x n p n = � i x i p i (where x i are the realizations, and p i are the weights) 15 / 30

Features of Random Variables Expected value (population mean) ◮ It is a weighted average of all possible values of X . The weights are determined by pdf. ◮ Precisely, if X is discrete and can take values { x 1 , . . . , x n } with pdf p i = P ( X = x i ), then the expected value of X is E ( X ) = x 1 p 1 + · · · + x n p n = � i x i p i (where x i are the realizations, and p i are the weights) 15 / 30

Example Question: X = { 4 , 12 , 2 , 6 } , P ( X = 4) = 0 . 1, P ( X = 12) = 0 . 2, P ( X = 2) = 0 . 5, P ( X = 6) = 0 . 2, calculate E ( X ) and E ( X 2 ). Answer: E ( X ) = 4 × 0 . 1+12 × 0 . 2+2 × 0 . 5+6 × 0 . 2 = 0 . 4+2 . 4+1+1 . 2 = 5. E ( X 2 ) = 4 2 × 0 . 1 + 12 2 × 0 . 2 + 2 2 × 0 . 5 + 6 2 × 0 . 2 = 1 . 6 + 28 . 8 + 2 + 7 . 2 = 39 . 6. 16 / 30

Example Question: X = { 4 , 12 , 2 , 6 } , P ( X = 4) = 0 . 1, P ( X = 12) = 0 . 2, P ( X = 2) = 0 . 5, P ( X = 6) = 0 . 2, calculate E ( X ) and E ( X 2 ). Answer: E ( X ) = 4 × 0 . 1+12 × 0 . 2+2 × 0 . 5+6 × 0 . 2 = 0 . 4+2 . 4+1+1 . 2 = 5. E ( X 2 ) = 4 2 × 0 . 1 + 12 2 × 0 . 2 + 2 2 × 0 . 5 + 6 2 × 0 . 2 = 1 . 6 + 28 . 8 + 2 + 7 . 2 = 39 . 6. 16 / 30