The random variable $X$ has a binomial distribution $\mathrm{B}\left( n,p \right)$, where $0 
$\frac{\mathrm{P}\left( X=r \right)}{\mathrm{P}\left( X=r-1 \right)}=\left( \frac{n-r+1}{r} \right)\left( \frac{p}{1-p} \right)$.[3]
Hence find a condition relating $n$ and $p$ such that $X$ has two values for its mode, and determine these two values, giving your answer in terms of $n$ and $p$.[3]

The random variable $X$ has a binomial distribution $\mathrm{B}\left( n,p \right)$, where $0 < p < 1$, and $n$ is an integer. Show that

$\frac{\mathrm{P}\left( X=r \right)}{\mathrm{P}\left( X=r-1 \right)}=\left( \frac{n-r+1}{r} \right)\left( \frac{p}{1-p} \right)$.

Hence find a condition relating $n$ and $p$ such that $X$ has two values for its mode, and determine these two values, giving your answer in terms of $n$ and $p$.

2016 HCI P2 Q6

2016 HCI P2 Q6

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