# Overview The probability section of this guide will likely never be fully completed, due to the fact that the [Prob 140 textbook](http://prob140.org/textbook/content/README.html) 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: * [**Counting**](https://cs70.bencuan.me/probability/counting) provides us an intuitive method of figuring out how many possible ways there are to do something. * [**Discrete probability distributions**](https://cs70.bencuan.me/probability/discrete-probability), such as the Binomial or Geometric distributions, describe the probabilities of a finite set of outcomes. * [**Continuous probability distributions**](https://cs70.bencuan.me/probability/continuous-probability), such as the Poisson or Normal distributions, help us model real values, like lifetime or height. * [**Markov chains**](https://cs70.bencuan.me/probability/markov-chains) model transitions between discrete states. * [**Expectation and variance**](https://cs70.bencuan.me/probability/expectation-and-variance) are tools to describe the characteristics of a random variable or distribution. * [**Concentration inequalities**](https://cs70.bencuan.me/probability/concentration-inequalities) allow us to approximate bounds for random variables when we only know their expectation and/or variance. 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] | $$\frac{1}{n-m+1}$$ | $$\frac{m+n}{2}$$ | $$\frac{(n-m+1)^2-1}{12}$$ | | |

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 $$\Gamma(r) = \int\_0^\infty x^{r-1}e^{-x}dx = (r-1)\Gamma(r-1) = (r-1)!$$