A book has $p$ pages and contains $q$ errors. The random variable $X$ is defined as the number of errors in a given page.
- What probability distribution law does $X$ follows and why?
- What is its expected value, its variance and its standard deviation?
Best Answer
You do have to make an assumption about how errors appear on a page.
If you assume that each of the $q$ errors has an equal probability of appearing on each page and that the page each error appears on is independent of the pages where other errors appear then given $p$ and $q$, for each page the probability distribution is binomial, so $$\Pr(X=x)= {q \choose x}\left(\frac{1}{p}\right)^x \left(1-\frac{1}{p}\right)^{q-x} = {q \choose x}\frac{\left(p-1\right)^{q-x}}{p^q}$$ with mean $\frac{q}{p}$, variance $\frac{q(p-1)}{p^2}$ and standard deviation $\sqrt{\frac{q(p-1)}{p^2}}.$