If X has Binomial distribution with parameters n and p such that np =λ, then \(\mathop {\lim }\limits_{n \to \infty } b\left( {x,n,p} \right);x = 0,1,2,.....\) is equal to:
With \(np=\lambda\), put \(p=\lambda/n\) in the Binomial PMF:
\[b(x,n,p)=\binom{n}{x}\left(\tfrac{\lambda}{n}\right)^x\left(1-\tfrac{\lambda}{n}\right)^{n-x}\]
Rearrange and take \(n\to\infty\): the factor \(\dfrac{n(n-1)\cdots(n-x+1)}{n^x}\to 1\), \(\left(1-\tfrac{\lambda}{n}\right)^{n}\to e^{-\lambda}\), and \(\left(1-\tfrac{\lambda}{n}\right)^{-x}\to 1\).
Therefore
\[\lim_{n\to\infty}b(x,n,p)=\dfrac{e^{-\lambda}\lambda^{x}}{x!},\quad x=0,1,2,\dots\]
This is the Poisson PMF, matching option (1).
Important Questions from Binomial Distribution
Indicate the correct answer for the combination from the following regarding the conditions for the applicability of a binominal distribution:
(a) There are n independent trials
(b) Each trial has only two possible outcomes
(c) The probabilities of two outcomes do not remain constant
(d) The trials are independent
Which of the following options is correct?
In which of the following practical situations, Poisson Distribution can be used?
A. Number of customers arriving at the super markets per hour.
B. Number of typographical errors per page in a typed material.
C. Number of accidents taking place per day on a busy road.
D. Dice throwing problems.
E. Number of defective material say, blades, etc. in a packing manufactured by a good concern.
Choose the most appropriate answer from the options given below:
For Binomial distribution, n = 10 and p = 0.6, E(X 2) (second moment about origin) is:
The mean and variance of binomial distribution B (x, n, p) are 4 and \(\dfrac{4}{3}\) respectively. What is the probability of getting 2 successes?
Find out the fallacy if any in the statement:
“The mean and the variance of a binomial distribution is 16.2 and 29.4 respectively.”
