Why Learn About a population?

Example: When Learning is Costly

• Maybe you are a hedge fund that wants to identify who might be a good trader. 

• Hire everyone that applies

• One Idea

• Allow everyone you hired to trade for one year

• Fire everyone who is not good at trading 

• Evaluate everyone's trading performance

Why is this not a good idea?

• Letting people trade who are not good at trading is expensive

• The avg. severance pay at Meta for the 11,000 employees laid of in 2022 was $88,000

• Initially, hire a small random subset of people who apply and evaluate them in a simulated trading environment

• Allow everyone you hired to trade

• In the future, hire only those who have a high estimated likelihood of real world trading success

• Learn the relationship between individual characteristics (like their score in the simulated trading environment) and real-world trading success

What's a Better Idea?

Why is this a Better Idea?

Probability Spaces

Motivation

• Composition is a fundamental way in which we can build new ideas using our existing ideas

• By learning about probability spaces, ideas in statistics become composable, which means we can build with them

Probability Space

• A probability space is a mathematical structure for representing uncertainty

• One can think of it as a tule with three elements

The set of all possible outcomes

The set of all events that we can assign probability to

Sample Space

Event Space

A Function that assigns the probabilities

Probability Measure

Probability Space Example

• Let's say that we are considering hiring some unknown individual to be a trader, how can we use a probability space to represent our uncertainty?

The set of all people we could hire (finite set)

The set of all subsets of people that we could hire

Sample Space

Event Space

A Function that assigns 1/n

probability to each {i}

Probability Measure

Probability Space Example (Continued)

• Our hedge fund doesn't care about the exact person that we hire 

• We don't care if the person is a Chicago Cubs fan

• We don't care if that person has a dog

• What we care about is the performance of the person as a trader. That is, how much $ they will generate for the firm

Probability Space Example (Continued)

• Our hedge fund doesn't care about the exact person that we hire 

• We don't care if the person is a Chicago Cubs fan

• We don't care if that person has a dog

• What we care about is the performance of the person as a trader. That is, how much $ they will generate for the firm