VITEEE 2018 Paper

The binomial distribution of a random variable $X$ is given by, $P(X=k)={}^{n}C_{k} p^{k} q^{n-k}$ Mean $= np$ Variance $= npq$ Given, the mean and variance of a random variable $X$ having a binomial distribution are 4 and 2 respectively, ⇒mean $= np = 4$ $\Rightarrow \text{ variance } = npq = 2$ From equation (1) and (2), we have $\Rightarrow q = \frac{1}{2}$ We know, $p = 1 - q$ $\Rightarrow p = \frac{1}{2}$ From equation (1), we have $n = 8$ The binomial distribution of a random variable $X$ is given by, $P(X=k)={}^{n}C_{k} p^{k} q^{n-k}$ $\Rightarrow P(X=1)={}^{8}C_{1} p^{1} q^{8-1}$ $\Rightarrow P(X=1)=8 \cdot \frac{1}{2} \cdot \frac{1}{2^{7}}$ $\Rightarrow P(X=1)=\frac{1}{32}$