CS70 Guide
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Overview

The probability section of this guide will likely never be fully completed, due to the fact that the Prob 140 textbook is such an excellent resource in probability theory. Go read it and do the problems!
Instead of a full write-up, the pages in this section will typically just link to relevant sections from the textbook. Personally, I found everything I needed to do well in CS70 probability (and much more) here, including examples that are very similar to problems you might see on the homework.
TL;DR don't use this section of the guide, just read the 140 textbook.
Here is a running list of topics in this section:
There is far more to explore in learning the basics of probability- not everything is included in this list!

Reference

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)!