Theory

Theory 1

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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.
• $P_X(k)={\displaystyle \left(\genfrac{}{}{0px}{}{n}{k}\right)p^k{q}^{n-k}}$
• $E[X]=np$
• $\mathrm{Var}[X]=npq$
• These are $n$ times the Bernoulli numbers.

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.
• $P_X(k)={\displaystyle \left(\genfrac{}{}{0px}{}{k-1}{\ell -1}\right)q^{k-\ell }{p}^{\ell }}$
• $E[X]={\displaystyle \frac{\ell }{p}}$
• $\mathrm{Var}[X]={\displaystyle \frac{\ell q}{p^2}}$
• These are $\ell$ times the Geometric numbers.

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

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

Theory 2

Uniform: $X\sim \mathrm{Unif}([a,b])$

• All times $a\le t\le b$ equally likely.
• $f_X(t)={\displaystyle \frac{1}{b-a}}$
• $E[X]={\displaystyle \frac{a+b}{2}}$
• $\mathrm{Var}[X]={\displaystyle \frac{1}{12}}{(b-a)}^2$

Exponential: $X\sim \mathrm{Exp}(\lambda )$

• Measures wait time until first arrival.
• $f_X(t)=\lambda e^{-\lambda t}$
• $E[X]={\displaystyle \frac{1}{\lambda }}$
• $\mathrm{Var}[X]={\displaystyle \frac{1}{{\lambda }^2}}$

Erlang: $X\sim \mathrm{Erlang}(\ell ,\lambda )$

• Measures wait time until ${\ell }^{\text{th}}$ arrival.
• $f_X(t)={\displaystyle \frac{{\lambda }^{\ell }}{(\ell -1)!}}{t}^{\ell -1}{e}^{-\lambda t}$
• $E[X]={\displaystyle \frac{\ell }{\lambda }}$
• $\mathrm{Var}[X]={\displaystyle \frac{\ell }{{\lambda }^2}}$

Normal: $X\sim 𝒩(\mu ,{\sigma }^2)$

• Limiting distribution of large sums.
• $f_X(x)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{e}^{-{(x-\mu )}^2/2{\sigma }^2}$
• $E[X]=\mu$
• $\mathrm{Var}[X]={\sigma }^2$