Probability (continued) Summary

•  P {Occurrence of A given that B has occurred}

= P (A|B)

Total probability:

• 

Baye's theorem:

• Let S be the simple space and E₁, E₂, E₃,….,E_n be n mutually exclusive and exhaustive events associated with a random experiment. If A is any arbitrary event then

Random Variable:

•  random variable is a function which associates a real number to each outcome of a random experiment.

Discrete random variable:

•  Random variables which can assume only a finite number, or a countable infinity number of values, are called discrete random variables.

Continuous random variable:

•  random variables which can assume any value over an interval.

Probability and Distribution

• If x is a discrete random variable assuming the values x₁, x₂, x₃,….,x_n with probabilities p₁, p₂, p₃,…., p_n respectively then (x₁,p₁), (x₂, p₂),…(x_n, p_n) defines a probability distribution of X.
•  Mathematical Expectation of X or Mean of X is

E (X) = x₁p₁ + x₂p₂ + ….+ x_np_n

Variance of X:

•  If X is a discrete random variable then variance of X denoted by V(X) is defined as

V(X) = E [X - E(X)]²

•  V(X) = E(X²) - [E(X)]²

Bernoulli trial:

• A random experiment which has only two outcomes (success or failure)
• Let n independent bernoulli trials be performed and X denote the number of successes in n trials then X follows a binomial distribution.
•  The probability of success be p,

The probability of failure = 1- p = q

The probability of r success in n trials

= P {X = r} = nC_r p^r q^n-r is called the probability mass function of the binomial distribution.

• Mean of binomial distribution = E(X) = np
• Variance of binomial distribution = V(X) = npq
• Poisson distribution is a limiting case of binomial distribution when n is very large, p is very small and np is a constant denoted (the parameter)
• Probability mass function of a poisson distribution is P(X = x)
• Mean of poisson distribution E(X) = l
• Variance of poisson distribution V(X) = l