Bayes Theorem, Binomial and Poisson Distributions

Introduction

     Suppose the two events are not independent, that is the occurrence of one depends on the occurrence of other, then how do we compute This can be explained by conditional probability.

     Baye's theorem is named after the British mathematician Thomas Bayes who published it in a research paper in 1763. It gives one of the important applications of the conditional probabilities by using the additional information supplied by the experiment or the past records.

Conditional Probability

     Let A and B be any two events associated with a random experiment. The probability of occurrence of event A when the event B has already occurred is called the conditional probability of A when B is given and is denoted as P(A/B).

Baye's Theorem

     Law of Total Probability: If B₁, B₂, B₃, ….., B_n are mutually exclusive and exhaustive events of the sample space S, then for any event A of S.

     Baye's Theorem: Let S be a sample space. If A₁, A, A₃ ... A_n are mutually exclusive and exhaustive events such that P(A_i) ? 0 for all i. Then for any event A which is a subset of

We have,

Random Variable and Probability Distribution

      Let S be a sample space associated with a given random experiment.

     A real valued function X which assigns to each  w_i Î S, a unique real number, X(w_i) = x_i is called a random variable. Two types of random variables are 1. Continuous random variable, 2. discrete random variable.

     Let X be a discrete random variable which takes values x₁, x₂, x₃,…x_n where p_i = P{X = x_i}

     Then X : x₁ x₂ x₃ …x_n

      P(X) : p₁ p₂ p₃ … p_n is called the probability distribution of x.

Binomial Distribution

     A trial, which has only two outcomes i.e., "a success" or "a failure", is called a Bernoulli trial.

      The probability distribution of the number of successes, so obtained is called the binomial distribution.

Poisson Distribution

     Poisson distribution is a limiting process of binomial distribution. Poisson distribution occurs when there are events which do not occur as outcomes of a definite number of outcomes.

     Poisson distribution is used under the following conditions:

     1. Number of trials n tends to infinity

     2. Probability of success p tends to zero and

     3. np = l is finite.

Applications

     Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation where it is called "pathway analysis".

     Example: In a housing colony 70% of the houses are well planned and 60% of the houses are well planned and well built. Find the probability that an arbitrarily chosen house in this colony is well built given that it is well planned.

Summary

     1. 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.

     2. Let n independent bernoulli trials be performed and X denote the number of successes in n trials then X follows a binomial distribution.

     3. 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.

     4. Mean of binomial distribution = E(X) = np.

     5. Variance of binomial distribution = V(X) = npq.

Conclusion

     In this chapter we have studied the method of evaluating probabilities of events relating to independent events and conditional events.

     We have also studied about random variables and their probability distributions, namely binomial distribution and Poisson distribution.