The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n\geq2$ a

Question

The random variable $X$ has the binomial distribution $\mathrm{B}(n,p)$, where $n\geq2$ and $0<p<1$. The mean of $X$ is three times its standard deviation. Show that $p=\frac{9}{n+9}$.[2] Given that $X$ is bimodal and $n<40$, find all possible values of $n$ and the corresponding modes of $X$.[4]

Accepted Answer

(a) $p=\frac{9}{n+9}$ (b) $n=3$: modes $2,3$; $n=9$: modes $4,5$; $n=15$: modes $5,6$; $n=27$: modes $6,7$

Two modes under a mean-standard-deviation condition

Two modes under a mean-standard-deviation condition

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