Probability Theory

Definition

  • Experiment can produce a number of outcomes (eg. roll dice with 6 possible outcomes)
  • A set of all outcomes is called the sample space of the experiment (ie. omega).
  • The power set of the sample space is formed by considering all different collections of possible results. (eg. P(odd)=> {1,3,5} which is subset of the power set).
  • ower set of the sample space.
  • Probability is a way of assigning every "event" a value between zero and one

Axioms

  • Non-negativity: P(A) >= 0
  • Addicitivity
  • Normalization

Random Variable

  • According to Khan, random variable is the variable that we assign it a value out from quantifying a random process. For example, X = 0 if coin flip shows head and X = 1 if it shows tail.
  • It is different from regular variable as it is the variable for probability.
  • You can have discrete and continuous random variables. Discrete is an distinct value while continuous with value like weight, height etc. But even discrete, you may have infinite number of values like a list of all odd years. Even we use discrete value to represent years, there are infinite numbers of years that fit.

Discrete vs Continuous probability distribution

Conditional Probability

  • The conditional probability of an event A is the probability that the event will occur given the knowledge that an event B has already occurred. This probability is written P(A|B), notation for the probability of A given B.
  • In the case where events A and B are independent, P(A|B) = P(A)
P(A|B) = P(A n B)/P(B)