Probability

• Axiomatic definition

  • axiom 1: p(A) ≥ 0;
  • axiom 2: p(S) = 1;
  • axiom 3: P(A1 ∪ A2 ∪ A3 ∪···) = P(A1) + P(A2) + P(A3) + ··· .
    if A _i are mutually exclusive
  • Consequences

    1. P(ϕ) = 0
    2. P(A1 ∪ A2 ∪ A3 ∪···) = P(A1) + P(A2) + P(A3) + ··· .
    3. if A ⊂ B then P(A) ≤ P(B)
    4. if A ∈ S then 0 ≤ P(A) ≤ 1;
    5. P(Aʹ) = 1 - P(A)
• Addition and Multiplication rule

  1. Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A∩B)
  2. Conditional Probability: P(A / B) = P(A∩B) / P(B)
  3. Multiplication Rule: P(A∩B) = P(A/B)* P(A / B)
• Theorem of Total Probability

  Total Probability:
• Baye's Theorem

  

Random Variables & Probability Distributions

• Random Variables

  Discrete Sample Space Continous Sample Space
  Probability Distribution Function(PDF):

  X	.	.	.	.	.	.	.	.
  f(X)	.	.	.	.	.	.	.	.

  Here 'X' is Random Variable and f(X) if Probability distribution function

  

  Cumulative Distrubution Function(CDF):

  

• Expectations

• Variences

• Types of distributions

  • Discrete Distributions
    • Binomial Distribution
    • Poisson’s Distribution
  • Continuous Distribution
    • Normal Distribution
    • Exponential Distribution

• Some other Distributions

  • Gamma Distribution
  • Beta Distribution

Sampling Distribution /Statistical Inferences

• Inferential statistics
• Central Limit Theorem

Testing of Hypothesis:

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• Scope of Hypothesis testing

• Procedures involved in hypothesis testing

• Applications

  • Types of Hypothesis & formulation

  • Errors in decision making

Random Process/Stohastic Process

• Markov Property
• Markov Chain n-step transition Probability
• Steady State Theorem

Queuing Theory

• Structure and component of Queuing System

  • Calling Population
  • Queuing Process
  • Queue Discipline
  • Service Process

• Kendall’s Notation

• The Birth and Death Process