Question

4 Sampling, Scoring, and Precision 96 8. Verify by calculation the following rejection technique for sampling values of x distributed. according to the Gaussian PDF [Kahn 1956]: f(x) = /2/7 exp[-2/2], for all x 2 0. (a) Select random numbers p, and p2 and let x -In pi and y = - In p2. (b) Accept xif 0.5(x-1)2 y;otherwise, return to step a. (c) Select random numbers p3 and assign to x thee negative sign if p3 < 0.5. Verify that the efficiency is given by /2 expl-1/2] 0.76.

Exploring monte carlo method example

Answer 1

I have generated random numbers p1, p2 , p3 in the range of 0 to 1. I did this exercise in excel and since I can not attach excel sheets, I gave a detailed explanation of how to get the efficiency number in excel below.

Step 1: Generating random sample of x and y values

Generate random number p1 and p2 between 0 and 1, by using the function rand()

Next get x and y values as -ln(p1) and -ln(p2) respectively. Excel has 'ln' function to get x and y values

Step 2 : Validating the condition between sample X and Y values

We can go to next step if the following condition holds otherwise we consider this sampling as failure and  will repeat the first step to get new samples of x and y values.

Condition : y>=0.5*(x-1)^2

Step 3: Getting final X value

Generate random number p3. if p3 >0.5 then x value remains same otherwise assign x to the negative sign.

Step 4: Calculating efficiency

Efficiency is ratio of samples which satisfy the condition in step-2 and total number of sample generated .

Following are the formulas to generate the sample in excel:

p1	p1	x	y	condition	p3	final x value
=RAND()	=RAND()	=-LN(C5)	=-LN(D5)	=IF(F5>=0.5*(E5-1)^2,1,0)	=IF(G5=1,RAND(),-99)	=IF(G5=0,-99,IF(H5<=0.5,-1*E5,E5))

if the condition holds true it will be assigned value of 1 otherwise 0. if the condition does not hold true I'm assigning value of -99 to p3 and final x values otherwise I calculate the p3 and final x values as mentioned above.

After simulating for 1000 sample i get the count of final x values with not equal to -99 value (good samples), and divide it by 1000 (total number of samples) to get the efficiency. Final efficiency i got was 0.764