Commit fcf696c3 authored by Robert Ricci's avatar Robert Ricci

Snapshot of Lecture 16 notes

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\title{CS6963 Lecture \#16}
\author{Robert Ricci}
\date{March 25, 2014}
\1 From last time
\2 How is the
\2 Thanks to Junguk and Makito for the scripts!
\1 Today: Talking about different prob distributions
\2 You many run into the need to generate data in these distributions,
or to recognize them in data that you get
\2 MLE: Maximum likelihood estimator: estimate parameters, each distrib
has its own
\2 Don't memorize the formulas, just be familiar with the concepts so that
you can look them up when needed
\1 Discrete distributions
\1 Bernoulli
\2 Just 1 and 0
\2 \emph{Why discrete?}
\2 Probability of a 1 is $p$
\2 Mean is $p$
\2 Variance $p(1-p)$ at lowest when $p$ is 0 or 1
\2 \emph{Examples of things modeled by Bernoulli distribs?}
\1 Binomial
\2 Number of successes ($x$) in a sequence of $n$ Bernoulli trials
\2 \emph{Why discrete?}
\2 So it has both $p$ and $n$ as params
\2 Mean: $np$
\2 Var: $n$ times var of Bernoulli
\2 \emph{Examples of things modeled by it?}
\1 Geometric
\2 Number of trials up to and including first success
\2 Param is just $p$
\2 Mean is $1/p$
\2 Remember, only for independent events!
\2 \emph{Examples?}
\1 Negative binomial
\2 How many successes before $r$ failures
\2 Can invert success of course
\2 Now you have $p$ and $r$ as parameters
\2 Mean: $\frac{pr}{1-p}$
\2 \emph{What might you model with it?}
\1 Continuous distributions
\1 Uniform: All possibilities equally likely
\2 There is a discrete version of course too
\2 Params: $a$ to $b$
\2 Mean: $\frac{a+b}{2}$
\2 Usually generated, not measured
\1 Exponential
\2 XXX More
\2 Parameter $\lambda$ - inverse of mean
\3 Sometimes called rate, eg. time between arrivals
\2 Memoryless: eg. time between arrivals
\3 No other continuous distribution has this property
\3 This property makes analysis simple
\3 But you have to be sure it's true!
\2 \emph{Examples?}
\1 Tails: Can be on both sides
\2 Heavy-tailed: Not exponentially bounded
\2 Fat-tailed: usually in reference to normal
\2 Long-tailed: usually in reference to exponential
\2 Means ``unlikely'' things are actually more common than one might expect
\2 80/20 rule
\2 Long tail means ``light'' somewhere else
\1 Normal
\2 We've talked plenty about, remember that sum of iid variables tends
towards normal
\1 Lognormal
\2 Logarithms turn multiplication into addition: $\log xy = \log x + \log y$
\2 So, lognormal is like normal, but for products of idd variables
\2 Useful for things that accumulate by multiplication, for example errors
\2 \emph{examples?}
\1 Pareto
\2 Discrete equivalent is Zipf
\1 Self-similarity
\1 For next time: Read papers
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