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Probability, Statistics, nd Data Analysis

创建日期：2024-08-15 16:18:58

所属分类：数学mathematics

标签： Probability

Probability, Statistics, nd Data Analysis

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Tutorial 5 – Probability, Statistics, nd Data Analysis

This tutorial presents some of the MATLAB functions used in probability calculations, basic statistics,

and data analysis. The discrete Markov chain model is introduced and its stationary points identified using

eigenanalysis, linking linear algebra and this class of stochastic model. Based on this, the PageRank algorithm

is discussed, which underpins Google’s websearch engine. Importantly, reading data from and writing data to

a file i s d emonstrated. Curve fitting is introduced, as is the correlation coefficient.

For a more detailed treatment of statistics and and machine learning topics, consult The Elements of

Statistical Learning: Data Mining, Inference, and Prediction, by Hastie, Tibshirani and Friedman (2009, 2nd

Ed, Springer Series in Statistics):

http://statweb.stanford.edu/~tibs/ElemStatLearn/printings/ESLII_print10.pdf

5.1 Combinatorics and Counting

Probability calculations often involve counting combinations of objects, and the study of combinations of

objects is the realm of combinatorics. Many formulas in combinatorics are derived simply by counting the

number of objects in two different w ays a nd s etting t he r esults e qual t o e ach o ther. O ften t he resulting

proof is obtained by a “proof by words.” Formulas are not necessarily derived from manipulations of factorial

functions as some students might think. Some of MATLAB’s combinatorial functions are illustrated in this

section, which are likely familiar to you.

5.1.1 Permutations and combinations

A permutation is an ordered arrangement of the objects in set S. If |S| = n, then there are:

n(n ? 1)(n ? 2) . . . (2)(1) = n!

different p ermutations o f t hese n o bjects. A n e xample i s t he s ize o f t he s et o f d ifferent or ders th at a group

n of people could appear in a queue, or that labeled balls are drawn from a bucket.

For these types of counting problems, MATLAB provides a factorial function. However, it is generally

preferable (because it is quicker and can be used for much larger numbers), to use the gamma function to

calculate factorials:

Γ (n) =

∫ ∞

xn?1e?xdx

and use the fact that Γ (n+ 1) = n! for positive integer values of n. For example,

f a c t o r i a l (5)

gamma(6)

ans =

120

ans =

120

More generally, a k-permutation arises when we choose k objects in order from a set of n distinct objects.

The total number of different permutations of size k of these n objects is:

n(n? 1)(n? 2) . . . (n? k + 1) = n!(n? k)!

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