glitinosim3

2022-07-09

A random variable, X, is defined as X ~ Geo(p). I know the mode is 1 as it is the value of X with highest probability.
How do i show this? As this is a discrete R.V, can i be allowed to use Calculus?

Jayvion Tyler

Expert

The probability distribution, as you've set it out, is
$P\left(X=k\right)=\left(1-p{\right)}^{k-1}p\phantom{\rule{2em}{0ex}}k\in \mathbb{N}$
It should then be simple to show that for any integer $k>1$,
$\begin{array}{rl}P\left(X=1\right)& =p\\ & >\left(1-p{\right)}^{k-1}p\\ & =P\left(X=k\right)\end{array}$
provided $0, so that $0<1-p<1$ and $0<\left(1-p{\right)}^{k-1}<1$.

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