Probability and Statistics Revision
Probability and Statistics Revision MH3512
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This 5 page Class Notes was uploaded by Andre Sõstar on Sunday August 23, 2015. The Class Notes belongs to MH3512 at Nanyang Technological University taught by TBA in Summer 2015. Since its upload, it has received 91 views. For similar materials see STOCHASTIC PROCESSES in Applied Mathematics at Nanyang Technological University.
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Date Created: 08/23/15
MH3512 Stochastic Processes Prof Nicolas Privault 1 Revision Probability and Statistics 11 Probability Spaces and Events A probability space is an abstract set Q that contains the possible outcomes of a random experiment An event is a collection of outcomes which is represented by a subset of Q aalgebras are collections 9 of events that we consider and these are assumed to satisfy following conditions 1 M Q 2 V countable sequences An 6 g n 2 1 we have U721 An 6 g 3 AEQgtQAEQwhereQAwEQw A The collection of all events in Q is often denoted F Empty set D and full space Q are considered events but they are less important Empty set cor responds to no outcome or experiment and Q corresponds to any outcome that can occur Example Q 1 2 34 56 The event A 246 corresponds to the result of the experiment is an even number Taking the same Q J 2 QD246135 de nes a aalgebra on Q which corresponds to the knowledge of parity of an integer picked at random from 1 to 6 Page 1 MH3512 Stochastic Processes Prof Nicolas Privault 12 Probability Measures A probability measure is a mapping 1P 7 gt 0 1 that assigns a probability IPA 6 01 to any event A with the properties 1 IPQ 1 and 2 2021 whenever Ak m A 019 739 When the subsets A1 An of Q are disjoint we have IPA1LJLJA IPA1IPA and IPA U B IPA IPB if A F B D In the general case we can write IPA U B IPA IPBIPA D B Basically you have to take the intersection of to sets off when calculating union otherwise you will take it two times 13 Conditional Probabilities and Independence Given any two events A B C Q with IPB 7E 0 we call IPA D B 193 the probability of A given B or conditionally to B Note that if IPB 1then IPA D B IPA and IPAB IPA IPAB 2 Law of Total Probability1 If B1 B2 B3 is a partition of the sample space S then for any event A we have IPA Z A n 18 Z IPAB IB Two events A and B are said to be independent if 1PAll 1PM that gives IPA D B IPAIPB 1 http wwwprobabilitycoursecomchapterl 1 42totalprobabilityphp Page 2 MH3512 Stochastic Processes Prof Nicolas Privault 14 Probability Distributions The probability distribution of a random variable X Q gt R is the collection PX E A A measurable subset of R Two random variables X and Y are said to be independent under the probability P if their probability distributions satisfy PX E AY E B PX E APY E B for all subsets A and B of R The function f IRIR is called the density2 of the distribution of X if b lPa g X S b 2 fada ie the distribution of X is absolutely continuous Let X and Y be continuous random variables with joint distribution function F Xy1y If there exists a nonnegative function fXyay such that a 21 3334 gt IPX S 3Y S y 2 fXystdsdt For example The marginal density functions 3 0351331 032431 07 elsewhere ay Then 1 xdy 1 Ody f1r13 mwdy f2w fxygtdy 0 f1y fxydx 20133da f2y fxyda 010dx 2https enWikipediaorgWikiProbabilitydensityfunction Page 3 MH3512 Stochastic Processes Prof Nicolas Privault For different distributions refer http matthiasvallentinnet blog 2010 10 distdiscpng 15 Expectation of a Random Variable The Expected Value of a random variable X usually denoted IEX deter mines the value that X assumes on average When the random experiment is repeated inde nitely The possible values of the random variable are weighted by their probabilities as speci ed in the following de nition De nition If X is a discrete random variable With frequency function px the expected value of X denoted by EX is IEX Z discrete case 2 IEX 1 f1da continuous case Let Y be a discrete random variable With probability function py and gY be a realvalued function of Y Then the expected value of gY is given by Elgyl Z 9ypy Vy De nition If X is a random variable With expected value lElX the vari ance of X is VarX IE3X IE3X2 provided that the expectation exists The standard deviation of X is the square root of the variance Variance is de ned as the eztpected squared dis tance between X and lElX ngxm m EX2 199033 if X is discrete VarX fie 33 EX2 fmada if X is continuous 00 Page 4 MH3512 Stochastic Processes Prof Nicolas Privault The conditional expectation of X w gt N given an event A is de ned by IE3XA Z kIPX MA 130 Lemma 11 Given an event A such that IPA gt 0 we have 1 IE X A 2 IE X1 l 1 PM A The Law of Total Expectation If X is an integrable random variable and Y is any random variable not necessarily integrable on the same probability space then EX ElElX Yll ZElX AHMAD the expected value of the conditional expected value of X given Y is the same as the expected value of X The expectation of a random sum 221 X where XkkEN is a sequence of random variables and if Y is independent of X k ng this yields Y 00 Y Z Xk 2 E 2 X1 1321 n0 k1 IE My n 16 Probability Generating Functions Consider X Q gt N U I oo a discrete random variable possibly taking in nite values The probability generating function of X is the function OX 2 1 gt IR de ned by GX8 2 anlPX n IE8X1Xlt100 1 g s g 1 0 n The dericative G Xs with respect to s satis es 0341 x i kIPX k k0 provided IEX lt 00 By computing the second derivative we can nd the variance Page 5
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