Probability Concepts: Discrete & Continuous Variables, Joint & Conditional Probabilities, Study notes of Economic statistics

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Introduction to Basic Probability Concepts

Charlie Gibbons Economics 140

September 3, 2009

Outline

1 Probability Basics Joint Probabilities Conditional Probabilities Independence

2 Expectations Definition and properties Conditional Expectations

3 Dispersion Variance Covariance and Correlation

Preliminary definitions

We begin with a random variable X. If X takes a countable number of values, it is discrete; otherwise it is continuous. Ex: The outcome of a die roll is a discrete random variable, while an individual’s income is a continuous random variable.

The simplest and most intuitive way to calculate a probability is the probability mass function (PMF). The PMF is Pr(X = x) and is calculated for discrete random variables.

For rolling a die, we have

Pr(roll a 1, 2, or 3) =

Preliminary definitions

Define the cumulative distribution function (CDF), FX (x), as Pr(X ≤ x). Ex: The CDF in the die rolling example calculates the probability of rolling a number less than x:

FX (3) = Pr(X ≤ 3) = Pr(X = 1) + Pr(X = 2) + Pr(X = 3) =

Preliminary definitions

We saw that the PMF of a discrete random variable is Pr(X = x); thus the CDF is

FX (x) =

∑^ x

y=−∞

Pr(X = y).

Preliminary definitions

We saw that the PMF of a discrete random variable is Pr(X = x); thus the CDF is

FX (x) =

∑^ x

y=−∞

Pr(X = y).

A continuous random variable has a sample space with an uncountable number of outcomes. Here, the CDF is defined as

FX (x) =

∫ (^) x

−∞

fX (y)dy.

Joint Probabilities

Previously, we considered the distribution of a lone random variable. Now we will consider the joint distribution of two random variables.

Joint Probabilities

The joint cumulative distribution function (joint CDF), FX,Y (x, y), of the random variables X and Y is defined by

FX,Y (x, y) = Pr(X ≤ x and Y ≤ y)

∑^ x

s=−∞

∑^ y

t=−∞

fX,Y (s, t) ds dt

As with any CDF, FX,Y (x, y) must equal 1 as x and y go to infinity.

Joint Probabilities

What is FX,Y (2, 3)?

Joint Probabilities

What is FX,Y (2, 3)?

x, y 1 2 3 4 5 6 1 1,1 1,2 1,3 1,4 1,5 1, 2 2,1 2,2 2,3 2,4 2,5 2, 3 3,1 3,2 3,3 3,4 3,5 3, 4 4,1 4,2 4,3 4,4 4,5 4, 5 5,1 5,2 5,3 5,4 5,5 5, 6 6,1 6,2 6,3 6,4 6,5 6,

FX,Y (2, 3) =

Joint Probabilities

What is fX,Y (6, 5)?

Joint Probabilities

What is fX,Y (6, 5)?

x, y 1 2 3 4 5 6 1 1,1 1,2 1,3 1,4 1,5 1, 2 2,1 2,2 2,3 2,4 2,5 2, 3 3,1 3,2 3,3 3,4 3,5 3, 4 4,1 4,2 4,3 4,4 4,5 4, 5 5,1 5,2 5,3 5,4 5,5 5, 6 6,1 6,2 6,3 6,4 6,5 6,

fX,Y (6, 5) =

Joint Probabilities

Joint CDF of independent normals

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Density

Joint Probabilities

Joint PDF of independent normals

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