JNTU B.Tech III RELIABILITY ENGINEERING Previous Question Paper (August/September - 2021 - R18)

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event. The probability mass function is given by:

where λ is the average number of occurrences in the interval and k = 0, 1, 2, ...

The relationship with the binomial distribution is defined by the Poisson limit theorem. If a binomial distribution has parameters n (number of trials) and p (probability of success), the Poisson distribution serves as an approximation when n is large (n → ∞) and p is small (p → 0) such that the product np remains constant at λ. In this limit, the binomial distribution B(n, p) converges to the Poisson distribution P(λ).

For a binomial distribution with parameters n and p, the expected value E[X] is derived as follows:

The probability mass function is P(X = k) = C(n, k) * p^k * q^(n-k), where q = 1 - p.

By definition, E[X] = Σ [k * P(X = k)] for k = 0 to n:

Since the term for k=0 is zero, we start from k=1:

Let m = n-1 and j = k-1. As k goes from 1 to n, j goes from 0 to m:

By the Binomial Theorem, the summation equals (p + q)^m = 1^m = 1. Therefore: