Question

If a random variable X follows a binomial distribution with mean 3 and variance 3/2, find P (X ≤ 5).

Answer 1

$$\begin{gathered} \mathrm{Mean}\left(np\right)=3\mathrm{and}\mathrm{variance}\left(npq\right)=\frac{3}{2} \\ \therefore q=\frac{1}{2} \\ \text{and}p\mathit{}=1-\frac{1}{2} \\ n=\frac{\mathrm{Mean}}{p} \\ \Rightarrow n=6 \end{gathered}$$

$$\begin{gathered} \mathrm{Hence},\mathrm{the}\mathrm{distribution}\mathrm{is}\mathrm{given}\mathrm{by} \\ P(X=r)={}^{6}C_r{\left(\frac{1}{2}\right)}^r{\left(\frac{1}{2}\right)}^{6-r},r=0,1,2...6 \\ =\frac{{}^{6}C_r}{2^6} \\ \therefore P(X\le 5)=1-P(X=6) \\ =1-\frac{1}{64} \\ =\frac{63}{64} \end{gathered}$$