6. Probability

数学英语



周维祺

Probability

The branch of mathematics on governing laws of random events

The Sample Space

The set of all possible outcomes of an experiment

Events

Subsets of a sample space

Probability Space

A sample space with a probability measure

Frequentist vs Bayesian

  • Frequentist: probability is the limiting value of the number of successes in a sequence of trials
    Randomness comes from incompleteness of trials

  • Bayesian: probability is a quantitative measure of uncertainty
    Randomness is its statistical description

Conditional Probability

The probability that occurs given that has occured

Independence

Two events are independent
if their occcurences do not depened on each other

Random Variable

A real valued function on the sample space with certain properties

Example

  • A coin is tossed times, let be the event that its head side turns up and the event that its tail side turns up

  • The total occurences of the head side is a random variable with
    Domain: , all possible outcomes
    Range: , the count of occurences of the head side

Example

  • Roll a die (a dice), and check whether the result is odd

  • This a random variable with
    Domain: , all possible outcomes
    Range: , true, false

Example

  • A character on a game map uses a skill called 'blink' that teleports him to a random position of the map immediately. There is a treasure cave on the map in which one gets a loot.

  • The distance to T after using the skill is a random variable with
    Domain: all possible positions of the map
    Range: to maximal possible distance to

(Cumulative) Distribution

The distribution fuction of a random variable is

Discrete random variable

The range of is countable
Its probability mass function

Continuous random variable

The distribution of is differentiable
Its probability density function

Neither continuous nor discrete

Such random variables exist

Example

  • Select a baseline on the ground, toss a coin, if the head turns up, then spin a rod on the ground and measure the final angle between the rod and the baseline, the angle shall be between (inclusive) and (not inclusive)




  • It has a point mass at

Game

  • You have the opportunity bet 1 on a number from 1 to 6

  • Three dice will be rolled, if your number failed to appear you lose, otherwise you get 1+x (so you win x) back where x is the occurence of your number

Game

  • play and discuss

  • is this game in your favour?

  • what if we replace x with 3 and 5 for pairs and three of a kind respectively?

Expectation (Mean)

Moments and Central Moments

Variance and Standard Deviation

Correlation

are uncorrelated if

otherwise they are correlated

i.i.d.

independently and identically distributed

Bernoulli Variable

A random variable that takes only two values

Example: Toss a coin, the head turns up with probability

Distributions

  • binomal distribution

  • geometric distribution

  • Poisson distribution

  • uniform Distribution

  • Normal Distribution

Gaussian

A random variable subject to the normal distribution

The Law of Large Numbers

Different mode (e.g., in probability, a.s.) of convergence of

as for i.i.d. variables with mean

Central Limit Theorem

Given i.i.d. variables with mean and variance ,

conveges in distribution to the standard Gaussian

Abstracts

Example

Writing Abstracts

  • Abstracts should be concise and precise, powerful and intriguing

  • What is achieved

  • How is it done
    optional: major difficulties

  • Why is it important
    optional: innovations, consequences

Tasks

Write a course paper, topics to be announced

Summary

  • Random variable, density/mass, distribution

  • Expection and moments

  • Central limit theorem

  • Writing abstracts