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

Question

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n$ is a positive integer and $0<p<1$. Denote $\mathrm{P}(X=r)$ by $q_r$. Show that $\frac{q_r}{q_{r-1}}=\frac{(n-r+1)p}{r(1-p)}$, for $r=1,2,\ldots,n$. Hence show that $X$ is bimodal if and only if $(n+1)p$ is an integer, and state its two modes in terms of $n$ and $p$.[4] It is known that the modes of $X$ are $11$ and $12$, and that $\mathrm{E}(X)=11.52$. Determine $n$ and $p$.[3]

Accepted Answer

(a) Bimodal if and only if $(n+1)p$ is an integer; modes $(n+1)p-1$ and $(n+1)p$ (b) $n=24$, $p=\frac{12}{25}$

Two modes from a probability ratio and mean

Two modes from a probability ratio and mean

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