By Finkenstadt B. F.
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Additional info for A stochastic model for extinction and recurrence of epidemics estimation and inference for measles o
12 Uniform distribution on 0; 1 . Let = 0; 1 and let F = B 0; 1 , the collection of all Borel subsets containined in 0; 1 . For each Borel set A 0; 1 , we define IP A = 0 A to be the Lebesgue measure of the set. Because 0 0; 1 = 1, this gives us a probability measure. This probability space corresponds to the random experiment of choosing a number from 0; 1 so that every number is “equally likely” to be chosen. Since there are infinitely mean numbers in 0; 1 , this requires that every number have probabilty zero of being chosen.
For every ! Z 2 X dIP = . 1 Z Yn dIP: CHAPTER 1. ; 0g; Z then we define X dIP = Z X + dIP Z , , X , dIP: If A is a set in F and X is a random variable, we define Z A Z X dIP = lIA X dIP: The expectation of a random variable X is defined to be IEX = Z X dIP: The above integral has all the linearity and comparison properties one would expect. In particular, if X and Y are random variables and c is a real constant, then Z Z If X ! Y ! for every ! Z X + Y dIP = 2 Z X dIP + Z Y dIP; cX dIP = c X dP; , then Z X dIP Z Y dIP: In fact, we don’t need to have X !
E, X ! e, 1 0 if ! if ! 2 A; 2 Ac ; X dIP = IP A: X ! ; where each ck is a real number and each Ak is a set in F , we define Z X dIP = Z n X k=1 ck lIAk dIP = n X k=1 ck IP Ak : If X is nonnegative but otherwise general, we define Z X dIP = sup Z Y dIP ; Y is simple and Y ! X ! for every ! 2 : In fact, we can always construct a sequence of simple functions Yn ; n = 1; 2; : : : such that 0 Y1 ! Y2 ! Y3 ! : : : for every ! 2 ; and Y ! 1 Yn ! for every ! Z 2 X dIP = .