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
Uniform(m,n)
[m, n]
$\frac{1}{n-m+1}$
$\frac{m+n}{2}$
$\frac{(n-m+1)^2-1}{12}$
Bernoulli(p)
Indicator
0, 1
P(X=1) = p
P(X=0) = 1-p
$p$
$p(1-p)$
Binomial(n,p)
[0, n]
$\binom{n}{k}p^kq^{n-k}$
$np$
$np(1-p)$
Poisson(
$\mu$
)
$x\ge0$
$e^{-\mu}\frac{\mu^k}{k!}$
$\mu$
$\mu$
Geometric(p)
$x \ge 1$
$(1-p)^kp$
$\frac{1}{p}$
$\frac{1-p}{p^2}$
Hypergeom.(N,G,n)
[0, n]
$\frac{\binom{G}{g}\binom{B}{b}}{\binom{N}{n}}$
$n\frac{G}{N}$
$n\frac{G}{N}\frac{B}{N}\frac{N-n}{N-1}$
Uniform Continuous
(a, b)
$\frac{1}{b-a}$
$\frac{a+b}{2}$
$\frac{(b-a)^2}{12}$
Beta(r,s)
(0, 1)
$\frac{\Gamma(r+s)}{\Gamma(r)\Gamma(s)}x^{r-1}(1-x)^{s-1}$
$\frac{r}{r+s}$
$\frac{rs}{(r+s)^2(r+s)}$
Exponential(
$\lambda$
)
(Gamma(1,
$\lambda$
))
$x\ge0$
$\lambda e^{-\lambda x}$
$\frac{1}{\lambda}$
$\frac{1}{\lambda^2}$
Gamma(r,
$\lambda$
)
$x\ge0$
$\frac{\lambda^r}{\Gamma(r)}x^{r-1}e^{\lambda x}$
$\frac{r}{\lambda}$
$\frac{r}{\lambda^2}$
Normal(0,1)
$x \in \mathbb{R}$
$\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$
0
1
Where
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
and
$\Gamma(r) = \int_0^\infty x^{r-1}e^{-x}dx = (r-1)\Gamma(r-1) = (r-1)!$