Theory

Theory 1

Bernoulli: $X\sim \mathrm{Ber}(p)$

• Indicates a win.
• $P_X(1)=p,P_X(0)=q$
• $E[X]=p$
• $\mathrm{Var}[X]=pq$

Binomial: $X\sim \mathrm{Bin}(n,p)$

• Counts number of wins.
• $P_X(k)={\displaystyle \left(\genfrac{}{}{0px}{}{n}{k}\right)p^k{q}^{n-k}}$
• $E[X]=np$
• $\mathrm{Var}[X]=npq$

Geometric: $X\sim \mathrm{Geom}(p)$

• Counts discrete wait time until first win.
• $P_X(k)=q^{k-1}p$
• $E[X]={\displaystyle \frac{1}{p}}$
• $\mathrm{Var}[X]={\displaystyle \frac{q}{p^2}}$

Pascal: $X\sim \mathrm{Pasc}(\ell ,p)$

• Counts discrete wait time until ${\ell }^{\text{th}}$ win.
• $P_X(k)={\displaystyle \left(\genfrac{}{}{0px}{}{k-1}{\ell -1}\right)q^{k-\ell }{p}^{\ell }}$
• $E[X]={\displaystyle \frac{\ell }{p}}$
• $\mathrm{Var}[X]={\displaystyle \frac{\ell q}{p^2}}$

Poisson: $X\sim \mathrm{Pois}(\lambda )$

• Counts “arrivals” during time interval.
• $P_X(k)={\displaystyle e^{-\lambda }\frac{{\lambda }^k}{k!}}$
• $E[X]=\lambda$
• $\mathrm{Var}[X]=\lambda$