EAS31116/B9036: Statistics in Earth & Atmospheric Sciences

Lecture 1: Review of ProbabilityInstructor: Prof. Johnny Luowww.sci.ccny.cuny.edu/~luo

Outlines

1. Definition of terms

2. Three Axioms of Probability

3. Some properties of probability

When facing uncertainties, we need a way to describe it. We can go with qualitative descriptors such as rain “likely”, “unlikely” or “possible”.

Probability is a quantitative way of expressing uncertainty, e.g., 40% chance of rain.

(Dictionary) Probability: the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.

Probability builds upon an abstract mathematical system.

Probability deals with uncertainties

Events: A set of possible (uncertain) outcomes (e.g., flipping coin: you won’t know for sure which face will come up).

Sample Space (or Event Space): the set of all possible events. Usually use capital letter S to represent it.

Mutually Exclusive and Collectively Exhaustive (MECE) events; ME: no more than one of the events can occur; CE: at least one of the events will occur.

A Few Terms

Venn Diagram

These are null space

Outlines

1. Definition of terms

2. Three Axioms of Probability

3. Some properties of probability

Axioms of Probability

• For an event E in a sample space S

exclusivemutually and where)()()( .3

1)( .2

10 .1

212121 EEEPEPEEP

SP

P(E)

Axiom: A self-evident truth that requires no proof; a universally accepted principle; (mathematics) a proposition that is assumed without proof for the sake of studying the consequences that follow from it.

The axioms are like the US Constitution. They are not very informative about what probability exactly means or how to estimate/interpret it.

There are two dominant views of the meaning of probability: the Frequency view and the Bayesian view.

Frequency view: The true probability of of event {E} exists and can be estimated through a long series of trials.

Bayesian view: There is no such a thing as true probability; we just estimate it based on whatever information we have in hand.

Here is how I would explain the basic difference to my grandma:

I have misplaced my phone somewhere in the home. I can use the phone locator on the base of the instrument to locate the phone and when I press the phone locator the phone starts beeping.

Problem: Which area of my home should I search?

Frequentist Reasoning: I can hear the phone beeping. I also have a mental model which helps me identify the area from which the sound is coming from. Therefore, upon hearing the beep, I infer the area of my home I must search to locate the phone.

Bayesian Reasoning:I can hear the phone beeping. Now, apart from a mental model which helps me identify the area from which the sound is coming from, I also know the locations where I have misplaced the phone in the past. So, I combine my inferences using the beeps and my prior information about the locations I have misplaced the phone in the past to identify an area I must search to locate the phone.

Outlines

1. Definition of terms & Venn Diagram

2. Three Axioms of Probability

3. Some properties of probability

Complement:

Complement:

Intersection (or joint probability)

Complement:

Union (one or the other, or both):

Intersection (or joint probability)

Complements of unions or intersections

Conditional Probability

Probability of an event, given that some other event has occurred or will occur.

For example, the probability of freezing rain, given the precipitation occurs.

Conditional Probability

Conditional probability can be defined in terms of the intersection of the events of interest and the condition event.

Independence

Two events are independent if the occurrence or non- occurrence of one does not affect the probability of the other.

Date (of Jan 1987); Precip (inch); T(max); T(min) in Ithaca NY

Estimate the probability of at least 0.01 in. of precipitation, given that T(min) is at least 00F.

(14/31)/(24/31) = 14/24 = 0.58

Estimate the probability of at least 0.01 in. of precipitation, given that T(min) is less than 00F.

(1/31)/(7/31) = 1/7 = 0.14

Think-Pair-Share: Why does wintertime precipitation prefer higher T?

Law of Total Probability

Why do we bother?

Sometimes we only know the conditional probability of {A} upon condition events Ei. The Law of Total Probability gives us an opportunity to estimate the unconditional probability Pr{A}

Ei are a set of MECE events

Bayes’ Theorem

So, if we know conditional probability Pr{E2|E1} and unconditional probability Pr{E1} and Pr{E2}, then we can back out Pr{E1|E2} .

Bayes’ Theorem

So, if we know conditional probability Pr{E2|E1} and unconditional probability Pr{E1} and Pr{E2}, then we can back out Pr{E1|E2} .

=

Date (of Jan 1987); Precip (inch); T(max); T(min) in Ithaca NY

In previous example, we estimate the conditional probability for precip occurrence given T(min) above or below 00F. Now, let’s use the Bayes’ Theorem to compute the converse conditional probabilities, concerning temperature events given that preci did or did not occur.