X B(7,p).

Question

$X\sim B(7,p)$. If $P(X=3)=5P(X=4)$, find the sum of the mean and variance of $X$.

Accepted Answer

${}^7C_3p^3q^4=5\cdot{}^7C_4p^4q^3 \Rightarrow \dfrac{{}^7C_3}{5\cdot{}^7C_4}=\dfrac{p}{q}\Rightarrow \dfrac{1-p}{1}=5p\Rightarrow p=\dfrac16,\ q=\dfrac56$.

$$\text{Mean}+\text{Var} = 7p+7pq = \frac76+\frac{35}{36}=\frac{42+35}{36}=\frac{77}{36}$$

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