Normal distribution is a limiting case of Binomial distribution under the following conditions.
(i) n, the number of trials is infinitely large i.e. n ⟶ ∞
(ii) Neither p(or q) is very small.

A random variable X is said to follow a normal distribution with parameters mean μ and variance σ², if its probability density function is given by

Poisson distribution is a limiting case of binomial distribution under the following conditions.
(i) n, the number of trials is indefinitely large ie., n⟶∞.
(ii) p, the constant probability of success in each trial is very small ie., p ⟶0.
(iii) np = λ is finite.Thus p = λ/n and q = 1 -(λ/n) where λ is a positive real number.

A random variable X is said to follow a Poisson distribution with parameter λ if it assumes only non-negative values and it probability mass function is given by
P(x,λ) = P(X = x)  {\(=\begin{cases} \begin{matrix} \frac { e^{-λ}{ \lambda }^{ x } }{ x! } , & x=0,1,2,....,\lambda >0 \\ 0, & otherwise \end{matrix} \end{cases}\)

Given mean of a binomial distribution is 5
np = 5 ....(1)
Also, standard deviation is 2 ⇒ Variance = 2² = 4
∴ npq = 4...(2)
(2) \(\div \) (1) given, \(\frac { npq }{ np } \)=\(\frac { 4 }{ 5 } \)
⇒ q = \(\frac { 4 }{ 5 } \)
p = 1-q = \(1-\frac { 4 }{ 5 } =\frac { 1 }{ 5 } \)
Substituting p = \(\frac { 1 }{ 5 } \) in (1) we get
n\(\left( \frac { 1 }{ 5 } \right) \) = 5 ⇒ n = 25
∴ The binomial distribution is ⁿC_x p^xq^n-x, x = 0,1,2,....n
⇒ ²⁵C_x \(\left( \frac { 1 }{ 5 } \right) ^{ x }\left( \frac { 4 }{ 5 } \right) ^{ 25-x }\), x = 0,1,2,....25.