\(U[a,b]\) |
\(\frac{1}{b-a+1},\, x=a,a+1,\dots,b\) |
Sample from \(\{a,a+1,\dots,b\}\) once uniformly at random\ |
\(Bin (n,p)\) |
\(\binom{n}{x}p^x(1-p)^{n-x},\,x=0,1,\dots,n\) |
\(\#\) of successes in \(n\) indep. trials with success prob. \(p\). |
\(Hyp(N,r,n)\) |
\(\frac{\binom{r}{x}\binom{N-r}{n-x}}{\binom{N}{n}},\) \(\max\{0, n-(N-r)\} \leq x \leq \min\{r,n\}\) |
\(\#\) of successes in \(n\) draws without replacement from \(N\) objects with \(r\) successes. |
\(NegBin(k,p)\) |
\(\binom{x+k-1}{x}p^k(1-p)^x,\, x=0,1,\dots,\) |
\(\#\) of failures until \(k\) successes in indep. trials with success prob. \(p\) |
\(Geo(p)\) |
\[p(1-p)^x,~ x=0,1,\dots \] |
\(\#\) of failures until first success in indep. trials with success prob. \(p\) |
\(Poi(\mu)\) |
\[\exp(-\mu) \mu^x/x!,~ x=0,1,\dots \] |
\(\#\) of occurrences in Poi process. |