The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$, where $n\ge 2$.

Question

The random variable $X$ has distribution $\mathrm{B}\left( n,p \right)$, where $n\ge 2$. It is known that the mean of $X$ and the standard deviation of $X$ are equal. Find an expression for $p$ in terms of $n$.[1] Show that $\frac{\mathrm{P}\left( X=1 \right)}{\mathrm{P}\left( X=2 \right)}=\frac{2n}{n-1}$. Hence find the the mode(s) of $X$.[4]

Accepted Answer

(a) ${p = \frac{1}{n+1}}$ (b) Mode: ${0}$ and ${1}$

2024 EJC P2 Q6

2024 EJC P2 Q6

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