CS70 Guide
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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

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)!
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Reference