The random variable $X$ is the number of successes in $n$ independent trials of an experiment in

Question

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$. Denoting $\mathrm{P}\left( X=k \right)$ by ${{p}_{k}}$, show that $\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{\left( n-k+1 \right)p}{k\left( 1-p \right)},k=1,2,...,n$. A distribution is said to be bimodal if it has two modes. Find the $3$ least values of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\frac{9}{16}$.

Accepted Answer

$\frac{p_k}{p_{k-1}}=\frac{(n-k+1)p}{k(1-p)}$ $n=15$: modes $8,9$ $n=31$: modes $17,18$ $n=47$: modes $26,27$

Prelims Revision Set: Bimodal Distribution Q2

Prelims Revision Set: Bimodal Distribution Q2

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