Class Note for MATH 115A with Professor Dawson at UA 2
Class Note for MATH 115A with Professor Dawson at UA 2
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Date Created: 02/06/15
te Carlo 1 Math 1 1 EA Spring 2008 Dawson Monte Carlo vs Bootstrapping o Bootstrapping uses computer simulation to form new samples from a large original sample Monte Carlo simulation creates samples from a known probability distribution All simulated values will also have the same distribution Recall We have looked at exponential distributions uniform distributions and binomial distributions Finite Random Variables Example As part of a drawing there is a box of 100 bills in the denominations of 1 5 and 10 The player will choose one bill out of the box Let X be the value of the bill ie X1 5 10 If the chosen bill is always replaced after each drawing the organizers would like to know how much they would expect to pay out after 100 drawings We will use the Excel functions RAND and IF to help in our simulation Finite Random Variables Example cont d Suppose the pmf for X is X 1 5 10 fXX 060 035 005 c We can compute the theoretical expected value EX EX1060503510005 060 175 050 285 0 Therefore if we do 100 drawings the expected total payout would be 100 times EX or 285 Finite Random Variables 0 Example cont d Using Excel 0 Use the RAND function to generate a random number with uniform distribution between 0 and 1 RAND is found under the Formulas Tab Math and Trig It doesn t require any input it generates a random number between 0 and 1 Use the IF function to generate the values of the drawings with the distribution of X IF is found under the Formulas Tab Logical Finite Random Variables Example cont d Using Excel u How should we use IF to make the proper distribution of X O 06 095 I I I I 1 5 1o 0 Therefore if RAND returns a value between 0 and 06 the IF function should return 1 between 06 and 95 the IF function should return 5 between 095 and 10 the IF function should return 10 Finite Random Variables 0 Example cont d o How could we do the same simulation using RANDBETWEEN instead of RAND Warning Computers simulate only your probability model not the actual real world situation 0 If you have a good model then the simulation will predict what is actually likely to happen If your initial numbers or probability assumptions are inaccurate then the simulation will not reflect the real world situation Continuous Random Variables In business applications it is often helpful to use a random variable s cdf to generate observations of the random variable The process of forming a new random sample is called the Monte Carlo method of simulation Continuous Random Variables o It becomes more complex to model continuous random variables by the distribution function If the variable was uniform over the interval 0 to 1 we could use RAND In the case of an exponential random variable it isn t that simple n We can t say generate a random number between 0 and infinity with an exponential distribution Exponential Random Variables RAND can still be useful RAND returns a randomly chosen number between 0 and 1 o For any random variable X o s PX s x s 1 c We can think of RAND returning the y value in the equation PX s x y and then we can solve for x Recall PX S X FXX Graphical Interpretation If we randomly chose a probability ie a number between 0 and 1 say 07 what would X have to be so that PXs X 07 Let a 2 Exponential Cd will occur when Xz 24 12 CHECK FX 24 167242 z 06988 FXQC Algebraic Interpretation The Cdf for an exponential random variable With a 2 is given by F 0 ifxlt0 Xx l e xZifXZO So if RAND selects 07 we are solving 07 1 e m 03267X2 1n03 x 2 x 21n03 z 2408 Algebraic Interpretation In general we are solving the following y 1eix0 eixZ i1n1 y a o We call this the inverse exponential Cdf Exponential Random Variables Example Use Excel to generate 300 random 9thptsheQatigns of an exponential random variable 0 Recall EX 0c 3 Monte Carlo Method By sampling from the distribution function we essentially have an infinite number of values to choose from o Those values will still have the same characteristics as the parent probability distribution same shape same mean o If sampling from a finite distribution then you will still have those same values of X and you will maintain the same proportion of them Objectives Give the definition of the term Monte Carlo simulation Use the IF and RAND functions in Excel to simulate observations of a random variable Use the IF and RANDBETWEEN functions in Excel to simulate observations of a random variable Use the inverse Cdf and the RAND function in Excel to generate random observations of an exponential random variable Use the ISNUMBER function in Excel
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