The skewness and kurtosis of a binomial distribution are $\frac{1}{6}$ and $-\frac{11}{36}$ respectively. Find the Binomial distribution.

1. Discrete distribution:

(i) Bernoulli's Distribution:

(a) Probability mass function of Bernoulli distribution is given by   $P(X=x)$$=\left\{\begin{array}{ll}{p}^x{q}^{1-x}& x=0,1\\ 0& otherwise\end{array}\right.$ where $q=1-p$.

(b) Here, mean$=p$, variance$=pq$ and standard deviation$=\sqrt{pq}$.

(ii) Binomial Distribution:

(a) A random variable $X$ denoting the number of successes in an outcome of a Binomial experiment having $n$ trials and $p$ as the probability of success in each trial is called Binomial random variable. Its probability mass function is given by

$P(X=x)$$=\{\begin{array}{cc}{}^{n}C_x{p}^x{q}^{n-x}& \text{for }x=0,1,2,\dots n\\ 0& \text{otherwise}\end{array}$ where $q=1-p$ and $n,p$ are its parameters.

(b) Here, mean$=np,$ variance$=npq,$ standard deviation$=\sqrt{npq}$ , skewness$=\frac{q-p}{\sqrt{npq}}$ and kurtosis$=\frac{1-6pq}{\sqrt{npq}}$.

(iii) Poisson Distribution:

(a) A random variable $X$ is said to follow a Poisson distribution if it assumes only non-negative integral values and its probability mass function is given by

$P(X=x)$$=\left\{\begin{array}{ll}\frac{e^{-\lambda }{\lambda }^x}{x!}& x=0,1,2,\dots \dots \\ 0& otherwise\end{array}\right.$, $\lambda$ is the parameter of the Poisson distribution.

(b) Here, mean $=\lambda =$variance, standard deviation $=\sqrt{\lambda }$, skewness $=\frac{1}{\sqrt{\lambda }}$ and kurtosis $=\frac{1}{\lambda }$.

2. Continuous Distribution:

(i) Regular or Uniform distribution:

(a) A random variable $X$ is said to have a continuous Uniform distribution over the interval $(a,b)$ if its probability density function is 

$f\left(x\right)$$=\left\{\begin{array}{ll}\frac{1}{b-a}& a<x<b\\ 0& \mathrm{otherwise}\end{array}\right.$. $a$ and $b$ are the parameters of uniform distribution.

(b) Here, mean $\mu =\frac{a+b}{2}$, variance ${\sigma }^2=\frac{(b-a{)}^2}{12}$, median$=\frac{a+b}{2}$, skewness$=0$ and kurtosis$=-\frac{6}{5}$.

(ii) Normal Distribution: 

(a) A random variable $X$ is said to have a Normal distribution with parameters $\mu$ (mean) and ${\sigma }^2$(variance) if its probability density function is given by $f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }} e^{-\frac{{\displaystyle 1}}{{\displaystyle 2}}{\left\{\frac{{\displaystyle x-\mu }}{{\displaystyle \sigma }}\right\}}^2}$ where $-\mathrm{\infty }<x<\mathrm{\infty },-\mathrm{\infty }<\mu <\mathrm{\infty }$ and $\sigma >0$. It is denoted by $X~N(\mu ,{\sigma }^2)$.

(b) Mean $=$ Mode $=$ Median $=\mu$, model height $=\frac{1}{\sigma \sqrt{2\mathrm{\pi }}}$, skewness ${\beta }_1=0$ and kurtosis ${\beta }_2=3$.

(c) Properties of Normal distribution: 

• $P(-\mathrm{\infty }<X<\mathrm{\infty })=1$
• $P(\mu -\sigma <X<\mu +\sigma )=0.6826$
• $P(\mu -2\sigma <X<\mu +2\sigma )=0.9544$
• $P(\mu -3\sigma <X<\mu +3\sigma )=0.9973$