Dr. Brajesh Kumar Jha

Poisson Distribution: Definition, Moments, MGF and Solved Examples Definition The Poisson Distribution is a discrete probability distribution used to model the number of times an event occurs in a fixed interval of time, space, area, or volume, when the events occur independently and at a constant average rate. A random variable \(X\) is said to follow a Poisson distribution with parameter \(\lambda > 0\) if \[ \boxed{ P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}, \qquad x=0,1,2,\ldots } \] We write \[ X\sim\operatorname{Poisson}(\lambda). \] Here, \(\lambda\) represents the average number of occurrences in the specified interval. Main Conditions Events occur independently. The average rate of occurrence is constant. The number of occurrences in disjoint intervals is independent. In a sufficiently small interval, the probability of more than one occurrence is negligible. Moments of the Poisson Distribution Let \[ X\sim\operatorname{Poisson}(\...

Binomial Distribution: Definition, Moments, MGF and Solved Examples A Binomial Distribution is a discrete probability distribution that gives the probability of obtaining exactly \(x\) successes in \(n\) independent Bernoulli trials, where the probability of success remains constant. If \(X \sim \operatorname{Bin}(n,p)\), then the probability mass function is \[ P(X=x)=\binom{n}{x}p^xq^{\,n-x}, \qquad x=0,1,2,\ldots,n, \] where \(n\) = number of trials, \(p\) = probability of success, \(q=1-p\) = probability of failure, \(X\) = number of successes. Conditions for Binomial Distribution The number of trials \(n\) is fixed. Each trial has only two possible outcomes: success or failure. The probability of success \(p\) remains constant. The trials are independent. Moments of Binomial Distribution Let \[ X\sim \operatorname{Bin}(n,p), \qquad q=1-p. \] First Moment: Mean \[ \boxed{E(X)=np} \] Second Raw Moment Since \[ E[X(X-1)]=n(n...