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 Robert Ricci committed Mar 25, 2014 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 \documentclass{article}[12pt] \usepackage[no-math]{fontspec} \usepackage{sectsty} \usepackage[margin=1.25in]{geometry} \usepackage{outlines} \setmainfont[Numbers=OldStyle,Ligatures=TeX]{Equity Text A} \setmonofont{Inconsolata} \newfontfamily\titlefont[Numbers=OldStyle,Ligatures=TeX]{Equity Caps A} \allsectionsfont{\titlefont} \title{CS6963 Lecture \#16} \author{Robert Ricci} \date{March 25, 2014} \begin{document} \maketitle \begin{outline} \1 From last time \2 Thanks to Junguk and Makito for the scripts!  Robert Ricci committed Mar 25, 2014 25 26  \2 Quick status: client, server, network conditions, client request sizes, analysis  Robert Ricci committed Mar 25, 2014 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67  \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?}  Robert Ricci committed Mar 25, 2014 68 69 70 71 72 73 74 75 76 77 78 79 80 \1 Poisson \2 The probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event'' \2 Produces a number of arrivals in a given time \2 Particularly good if the sources are independent \2 Parameter is mean ($\lambda$) \2 Very often used for arrivals: eg. arrival of packets at a queue or requests at a server \2 Can be used over particular intervals of time; eg. daytime, to keep the iid assumption \2 \emph{Examples?}  Robert Ricci committed Mar 25, 2014 81 82 83 84 85 86 87 88 89 \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  Robert Ricci committed Mar 25, 2014 90  \2 Models length of time between arrivals (compare to Poisson)  Robert Ricci committed Mar 25, 2014 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113  \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 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  Robert Ricci committed Mar 25, 2014 114  \2 \emph{Examples?}  Robert Ricci committed Mar 25, 2014 115 116  \1 Pareto  Robert Ricci committed Mar 25, 2014 117  \2 Produces IID inter-arrival times  Robert Ricci committed Mar 25, 2014 118  \2 Discrete equivalent is Zipf  Robert Ricci committed Mar 25, 2014 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137  \2 Power law: few account for the largest portion - eg. the 99%'' \2 Self-similar: the same at different scales (think fractal) \2 Bursty on many or all time scales'' \2 Values correlated with future incidents \2 Compare to other distributions\ldots eg Poisson \2 Can be constructed with heavy-tailed ON/OFF sources \2 Has either no mean or infinite variance \1 Weibull \2 Good for monitoring mean time to failure \2 Parameters are scal ($\lambda$) and shape ($k$) \2 A value of k < 1 indicates that the failure rate decreases over time'' \2 A value of k = 1 indicates that the failure rate is constant over time.'' \2 A value of k > 1 indicates that the failure rate increases with time.'' \1 For next time: \2 Read papers for Analysis 3 \2 Posted one reading for next Thursday \2 Lab due a week from today  Robert Ricci committed Mar 25, 2014 138 139 140 141  \end{outline} \end{document}