Overview

This page will be populated in preparation for Finals week (mid-December 2020). In the meantime, use the Prob 140 textbook as an excellent resource in probability theory.

Here is a running list of topics that are currently covered:

  • Counting provides us an intuitive method of figuring out how many possible ways there are to do something.

  • More TBA!

Reference

http://prob140.org/assets/final_reference_fa18.pdf

Distribution

Values

Density

Expectation

Variance

Links

Uniform(m,n)

[m, n]

1nm+1\frac{1}{n-m+1}

m+n2\frac{m+n}{2}

(nm+1)2112\frac{(n-m+1)^2-1}{12}

Bernoulli(p)

Indicator

0, 1

P(X=1) = p

P(X=0) = 1-p

pp

p(1p)p(1-p)

Binomial(n,p)

[0, n]

(nk)pkqnk\binom{n}{k}p^kq^{n-k}

npnp

np(1p)np(1-p)

Poisson(μ\mu)

x0x\ge0

eμμkk!e^{-\mu}\frac{\mu^k}{k!}

μ\mu

μ\mu

Geometric(p)

x1x \ge 1

(1p)kp(1-p)^kp

1p\frac{1}{p}

1pp2\frac{1-p}{p^2}

Hypergeom.(N,G,n)

[0, n]

(Gg)(Bb)(Nn)\frac{\binom{G}{g}\binom{B}{b}}{\binom{N}{n}}

nGNn\frac{G}{N}

nGNBNNnN1n\frac{G}{N}\frac{B}{N}\frac{N-n}{N-1}

Uniform Continuous

(a, b)

1ba\frac{1}{b-a}

a+b2\frac{a+b}{2}

(ba)212\frac{(b-a)^2}{12}

Beta(r,s)

(0, 1)

Γ(r+s)Γ(r)Γ(s)xr1(1x)s1\frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)}x^{r-1}(1-x)^{s-1}

rr+s\frac{r}{r+s}

rs(r+s)2(r+s)\frac{rs}{(r+s)^2(r+s)}

Exponential(λ\lambda)

(Gamma(1, λ\lambda))

x0x\ge0

λeλx\lambda e^{-\lambda x}

1λ\frac{1}{\lambda}

1λ2\frac{1}{\lambda^2}

Gamma(r, λ\lambda)

x0x\ge0

λrΓ(r)xr1eλx\frac{\lambda^r}{\Gamma(r)}x^{r-1}e^{\lambda x}

rλ\frac{r}{\lambda}

rλ2\frac{r}{\lambda^2}

Normal(0,1)

xRx \in \mathbb{R}

12πe12x2\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}

0

1

Where (nk)=n!k!(nk)!\binom{n}{k} = \frac{n!}{k!(n-k)!}and Γ(r)=0xr1exdx=(r1)Γ(r1)=(r1)!\Gamma(r) = \int_0^\infty x^{r-1}e^{-x}dx = (r-1)\Gamma(r-1) = (r-1)!