Question

If for a binomial distribution P (X = 1) = P (X = 2) = α, write P (X = 4) in terms of α.

Answer 1

$$\begin{gathered} \text{F}\mathrm{or}\mathrm{binomial}\mathrm{distribution}\mathrm{of}X, \\ P(X=r)={}^{n}C_r\left(p{)}^r\right(q{)}^{n-r},r=0,1,2,...,n \\ P(X=1)=np(q{)}^{n-1} \\ P(X=2)={}^{n}C_2{p}^2(q{)}^{n-2} \\ \Rightarrow np(q{)}^{n-1}={}^{n}C_2{p}^2(q{)}^{n-2}=\alpha \\ \mathrm{Simplifying}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{we}\text{get,} \\ q=\frac{n-1}{2}p \\ \Rightarrow 2q=np-p \\ \mathrm{On}\mathrm{putting},q=1-p\mathrm{we}\mathrm{get} \\ 2-2p=np-p \\ p(n+1)=2.....\left(\mathrm{i}\right) \\ \mathrm{Also},P(X=1)=\alpha \\ \Rightarrow np(1-p{)}^{n-1}=\alpha .....(\mathrm{ii}) \end{gathered}$$

Note: We cannot find the value of n as (i) and (ii) are not linear and hence we cannot find the value of P(X = 4)