Probability and Statistics Reviews, Part One

Classic Probability Problem - Standard Procedures

  1. Define simple events (basic events) based on the given information
  2. Abstract the problem to be solved into a compound event
  3. Solve the problem based on classic probability equations

Random Experiment

The concept of random experiment arise from the assumption that the event to be observed is described on a occurring or not basis. Many real world probability problem, however, does not self-describe in such nature. These types of problems can be conceptually transformed into the occurring&observation pattern.

Random Variable

A random variable is the mapping of the result of a random experiment to a real value.

Statistics and Probability

Statistics is the necessary measure that takes to reach a probability (when the population is known) or probability estimate (when only sampling is available).

Relative Frequency

Phrase "Relative" here refers to the fact that in any experiment that is infinitely repeatable, the absolute frequency (proportion ratio) cannot be obtained.

Simple Event (Elementary Event)

A random experiment with an outcome (coin toss with a head on top).

In contrast, compound events have multiple outcomes. An elementary event can be a process with outcomes that are "complex", for example, toss 3 dices and get 1, 3, 2 on top. In this case, the elementary event A here is constructed by three elementary events in a related probability space (toss only one dice to see the outcome).

Coin toss with a head on top and coin toss with a tail on top are two simple events.
Simple and compound events are relative concepts, when no lower level events are defined, a complex events may be defined as simple events when analyzing certain problems.

Sample Space (One Element of Probability Space)

A sample space is a set, denoted as S, which enumerates each and every possible outcome or simple events.
All possible simple events (event condition together with its outcomes).

Any events (to be observed) covers part of the sample space (the event space is a subset of sample space)

The equation of conditional probability should only be applied when P(A), P(B) and P(A*B) are reflects the probability of events under the same sample space.

Distribution Mixture

N1 = The population size of distribution 1
N2 = The population size of distribution 2
P3(x) = (P1(x)N1 + P2(x)N2)/(N1+N2)