If X ~ B(n, p), E(X) = 12, V(X) = 4, then find n - Vidyadip

Question

If X ~ B(n, p), E(X) = 12, V(X) = 4, then find n

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Answer

As $X \sim B(n, p), E(X)=n p$ and $V(X)=n p q$
It is given that $E(X)=12$ and $V(X)=4$
$ \therefore \frac{ V ( X )}{ E ( X )}=\frac{4}{12}$
$\therefore \frac{ npq }{ np }=\frac{1}{3}$
$\therefore q =\frac{1}{3}$
$\therefore p =1- q$
$=1-\frac{1}{3}$
$=\frac{2}{3}$
$\therefore E ( X )= np =12$
$\Rightarrow n \times \frac{2}{3}=12$
$n =18 $

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