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

Question

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

Accepted Answer

${X}$ will have two modes when ${(n+1)p}$ is a positive integer. ${r=(n+1)p}$ , ${r=(n+1)p-1}$

2016 HCI P2 Q6

2016 HCI P2 Q6

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