In a binomial distribution B(n,p), the sum and product of the mean and variance are 5 and …

Question

In a binomial distribution $B(n,p)$, the sum and product of the mean and variance are 5 and 6 respectively. Find $6(n+p-q)$.

Accepted Answer

$np+npq=5$, $np\cdot npq=6$. Solving: $q=\dfrac23$ or $\dfrac32$ (reject), so $q=\dfrac23,\ p=\dfrac13$. Then $n\cdot\dfrac13+n\cdot\dfrac13\cdot\dfrac23=5\Rightarrow\dfrac{3n+2n}{9}=5\Rightarrow n=9$.

$$6(n+p-q) = 6\left(9+\frac13-\frac23\right) = 6\left(9-\frac13\right)=54-2=52$$

Solutions · 1