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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
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Verified Answer
The correct answer is:
$\frac{1}{3}$
Here, $n=6$
According to the question
$\begin{array}{rlrl}
{ }^6 C_2 p^2 q^4=4 \cdot{ }^6 C_4 p^4 q^2 \\
\Rightarrow q^2 =4 p^2 \\
\Rightarrow (1-p)^2 & =4 p^2 \\
\Rightarrow 3 p^2+2 p-1 & =0 \\
\Rightarrow & (p+1)(3 p-1) & =0 \\
\Rightarrow & =\frac{1}{3} \\
& \because p \text { cannot be negative })
\end{array}$
According to the question
$\begin{array}{rlrl}
{ }^6 C_2 p^2 q^4=4 \cdot{ }^6 C_4 p^4 q^2 \\
\Rightarrow q^2 =4 p^2 \\
\Rightarrow (1-p)^2 & =4 p^2 \\
\Rightarrow 3 p^2+2 p-1 & =0 \\
\Rightarrow & (p+1)(3 p-1) & =0 \\
\Rightarrow & =\frac{1}{3} \\
& \because p \text { cannot be negative })
\end{array}$
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