What is probability?

• Probability is the science of theoretical & numerical representation, modeling, analysis & control over uncertainty, imperfection, variability, sampling, interdependence & cost-benefit factors of a particular event

Why probability?

• In any event, task, studies or phenomenon, one often focuses on a few selected components or factors that relate to one's needs or requirements
• If these factors are covered entirely within theoretical domains (like any tutorial exercise), stating & applying the appropriate principles would solve the problems, thus the chances or probability of a confident solution is 100%
• However, in the real-world scenario, the events might not be explained fully by theory or even if the theory is available, it might not be easily understood or readily available (expertise & technology)
• Thus, design, control & decision making under uncertainty that can be modeled by probability, the chances & stochastic details of the event

Uses?

• Uncertainty: weather data, hydrological routing, production line, construction planning
• Decisions: strategy, implications of findings, expectations, risk management, volatility
• Understanding: research, game theory, probabilistic relationships, prediction & forecasting

Basis?

• Probability of event I: 0 <= P(I) <= 1, from probabilistically impossible to certainty
• Definitions
• Set theory
• Venn diagrams: events, complementary, union & intersection
• Axioms: mutual exclusive, de Morgan's rule
• Conditional: P(A|B) & statistical independent: P(A.B)=P(A).P(B)
• Theorem of total probability: P(A)=sum of P(A|B).P(B)
• Bayes' theorem: P(A|Ei).P(Ei)=P(Ei|A).P(A)
• Discrete: PMF - P(X=xi) & CDF - F(x)=P(X<=x)
• Continuous: PDF - f(x) & P(x)=integral f(x).dx

What is X?

• X or any other random variable is a probabilistic representation of the event in question
• For example, X can be distance for traffic problem
• The major terms to describe random variable, rv:

1. Mean (formula), mode, median: weighted average
2. Standard deviation, variance, coefficient of variation (COV): measure of degree of dispersion
3. Moments: various powers of measure of dispersion, skewness, etc., from moment generating function

What are distributions?

• Sampling data may fit into known, theoretical spread of data curves called distributions
• These discrete & continuous functions exist due to assumed modeling, limiting process & statistics
• Continuous: Normal, Lognormal, Poisson, Exponential
• Discrete: binomial, geometric

Multiple R.V.?

• Marginal probability, Pxy(x,y) or fxy(x,y) is the chance of a particular (x,y) combination in the entire domain
• For more rv., expand the probability
• Bivariate distribution: distribution of 2 rv.

More concepts?

• Covariance, Cov(x,y): measure of linear interdependence between rv. x & y
• Correlation: by Schwarz's inequality, (-1,1)
• Functions of random variables:

1. Single variable: Y=g(X)
2. Multi-value: Y=g(X1, X2, …, Xn)
3. Multiple variables: Y1, Y2, … Ym

• Special cases: mean & variance of

1. Sum of Normals: Y=a + sum of ai.Xi
2. Product of Lognormals: ln Y=ln c+sum ai.ln Xi

• Chebyshev inequality: with mean & variance known, analyse other variables
• Law of large numbers: sample mean & variance tend towards those of population
• Central Limit Theorem: under general conditions, distribution of variables tends to Normal:

1. R.V.>30
2. No one dominates
3. Variables not highly dependent

• Consider statistics