# Lec16 - i=1 n Vi i=1 n Vi V 2 i.e E(V = V 1 1 N i=1 N Vi where 3 Vi = Xi i = 1 2 100 and Xi is the average of the ith sample The larger N is the

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Elementary Simulation Idea: Numerical simulation allows us to get approximate results to all kinds of probability problems, that can’t be solved or diﬃcult to solve analytically. Basic problem: 1. X,Y, ··· ,Z are independent random variables 2. g ( x, y, ,z ) , is an arbitrary function deFned on R k 3. V = is a random variable of interest, e.g. X Y or / (sin( Z )+1) etc We are interested in evaluating P (13 <V < 17) E ,Var etc. Unless is simple, is small, and we are LUCKY, we may not be able to solve these problem analytically or explicitly. How to obtain these via simulation? 1 Simulation Framework ±ramework of simulation: 1. Generate some large number (say n )ofvaluesforeachofthe variables . We then have a set of nk tuples of the form i ,Y ,i =1 ,n Plug each into function and evaluate for each tuple: 3. Then: (a) a b is approximated by ±V : [ a, b ] 2 (b) . e )= ¯ (c) Var Example: Estimate the expected value of the mean of a sample of size (say, 100) from an Erlang distribution Erlang (4 2) , i.e., want μ where + ... 100 and ,X ,...,X iid To do this using simulation, we need to genearate N samples each of size 100 from the distribution, calculate the average from each sample, then average them over the samples. i.e. ,..., istheaverageofthe th sample. The larger is, the more accurate this estimate as the standard error is reduced. 3 Random Number Generators All of you will be familiar with those questions from an IQ test, where you are supposed to Fnd the next number in a sequence. Possible sequences could be: 1. 1,3,5,7,9, . .. 2. 1,4,9,16, 25, . 3. 1,4,1,5,9,2,6, . 4. 4, 1, 5, 3, 9, . Can you guess the rules of them? 4

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Random Number Generators (Cont’d) The rules behind the frst two sequences are trivial The third one is tricky. OF course, there are infnite many possibilities to produce the same fnite stream oF numbers. The Fourth sequence was produced by the Function x i +1 =(2 +3) mod10 i.e. take the last number, multiply it by 2, add 3 and take only the last digit to get the next number in the sequence. The next number would thereFore
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## This note was uploaded on 02/01/2012 for the course STAT 330B taught by Professor Zhou during the Spring '11 term at Iowa State.

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Lec16 - i=1 n Vi i=1 n Vi V 2 i.e E(V = V 1 1 N i=1 N Vi where 3 Vi = Xi i = 1 2 100 and Xi is the average of the ith sample The larger N is the

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