Q57 / 120 · VITEEE 2009 · Mathematics

If $X$ is a binomial variate with the range $\{0,1,2,3,4$, $5,6\}$ and $P(X=2)=4 P(X=4)$, then the parameter p of $X$ is

  • Correct answer
    A
    $\frac{1}{3}$
  • B
    $\frac{1}{2}$
  • C
    $\frac{2}{3}$
  • D
    $\frac{3}{4}$

Solution

Here, $n=6$ According to the question $$ \begin{array}{l} { }^{6} C_{2} p^{2} q^{4}=4 \cdot{ }^{6} C_{4} p^{4} q^{2} \\ \Rightarrow q^{2}=4 p^{2} \Rightarrow(1-p)^{2}=4 p^{2} \end{array} $$ $$ \begin{array}{l} \Rightarrow 3 p^{2}+2 p-1=0 \\ \Rightarrow(p+1)(3 p-1)=0 \\ \Rightarrow p=\frac{1}{3} \quad(\because p \text { cannot be negative }) \end{array} $$

Hence, the correct answer is option: 'A'.