16 Lecture 15, Feburary 09, 2024

16.1 Review of the distributions we covered before

Distribution \(f(x)=P(X=x)\) Interpretation
\(U[a,b]\) \(\frac{1}{b-a+1},\, x=a,a+1,\dots,b\) 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.