\displaystyle \textbf{1.}\ \text{A random variable }X\text{ which takes values }0,1,2,\ldots,n\text{ is said to follow a} \\ \text{binomial distribution if its probability distribution function is given by}
\displaystyle \mathrm{P}(X=r)={}^{n}C_{r}p^{r}q^{\,n-r},\quad r=0,1,2,\ldots,n,\text{ where }p,q>0\text{ such that } \\ p+q=1.
\displaystyle \text{The two constants }n\text{ and }p\text{ in the distribution are known as the parameters of the} \\ \text{distribution.}
\displaystyle \text{The notation }X\sim B(n,p)\text{ is generally used to denote that the random variable } \\ X\text{ follows a binomial distribution with parameters }n\text{ and }p.
\displaystyle \text{We have,}
\displaystyle \mathrm{P}(X=0)+\mathrm{P}(X=1)+\cdots+\mathrm{P}(X=n)={}^{n}C_{0}p^{0}q^{n}+{}^{n}C_{1}p^{1}q^{n-1}+\cdots+{}^{n}C_{n}p^{n}q^{0}.
\displaystyle =(q+p)^{n}=1^{n}=1.
\displaystyle \text{Thus, the assignment of probabilities to the random variable }X\text{ is permissible.}

\displaystyle \textbf{2.}\ \text{If }n\text{ trials constitute an experiment and the experiment is repeated }\\ N\text{ times, then the frequencies of }0,1,2,\ldots,n\text{ successes are given by}
\displaystyle N\cdot\mathrm{P}(X=0),\;N\cdot\mathrm{P}(X=1),\;N\cdot\mathrm{P}(X=2),\ldots,N\cdot\mathrm{P}(X=n).

\displaystyle \textbf{3.}\ \text{The mean and variance of a binomial variate with parameters }n\text{ and }\\ p\text{ are }np\text{ and }npq\text{ respectively.}

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