Introduction to Probability and Statistics4th Week (3/29)

1. Bayer’s Theorem2. Random Variables

3. Probability Distributions4. Mathematical Expectations (intro)

 

What would you do…..

IF a medical test (tumor marker) inform you that you got an incurable disease (i.e. Pancreases Cancer)

1.Cry2.Use your remaining time for some important thing3.Invent a new iphone

Baye’s Theorem: Definition

Baye’s Theorem: Proof

• Why do we care?? • Why is Bayes’ Rule useful?? • It turns out that sometimes it is very useful to be able to

“flip” conditional probabilities. That is, we may know the probability of A given B, but the probability of B given A may not be obvious.

Baye’s Theorem: When do we need?

Baye’s Theorem: Example

Random Variables

Las Vegas

777(Jack Pot) => 1 million dollars (1)Others: Bam => 0 dollars (0)

How often do you get “1”?

How much do you put money to get 1 million dollars?

Discrete Probability Distributions

Discrete Probability Distributions

Distribution Function

Distribution Function for Discrete Random Variables

Distribution Function for Random Variable

Distribution Function for Discrete Random VariablesDistribution Function

Continuous Probability Distributions

Example

Example

Joint Distribution

Joint Distribution: An Example

X: Get A+ for P&S

Y: Get a great boy/girl friend

- Dependent? - Independent?

X

Y

Get a friend

No friend

A+ Others

Discrete Joint Probability Function

Discrete Joint Distribution Function

Probability Function (it’s like a point)

Distribution Function (it’s like an area)Understand the difference between

Continuous Joint Distribution Function/Distribution

Probability Surface

Probability Function

Marginal Distribution Function

We call them the marginal distribution functions, or simply the distribution functions, of X and Y, respectively.

Density Function

Independent Random Variables

Independent Random Variables

Changes of Variables

Changes of Variables

Changes of Variables: Example

Changes of Variables: Example

Probability Distributions of Functions of Random Variables

Convolutions

Conditional Distributions: Discrete

Conditional Distributions: Continuous

Conditional Distributions: Example

Applications to Geometric Probability

Mathematical Expectations*: Definition

*in Korean: 기대값

- Discrete

- Continuous

Mathematical Expectations: Example

Mathematical Expectations: Example

Functions of Random Variables

Functions of Random Variables

Functions of Random Variables

A Few Theorems on Expectation

The Variance and Standard Deviation

The Variance and Standard Deviation

The Variance and Standard Deviation

The Variance and Standard Deviation

A Few Theorems on Variance

Compare!

is true for any random variables

is true for only independent variables

is true for only independent variables

Not “Var(X) – Var(Y)”

Vs.

Standardized Random Variables