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STA 6106 – Fall 2009
Simulation of Two Independent Random Samples from a Standard Normal Distribution
Using the Box-Muller Transformation
The following program simulates two independent random samples of an arbitrary size from a standard
normal distribution, through use of the Box-Muller transformation. The input values are the sample
size and the initial seed.
5
15
PROGRAM BOXMULLER
INTEGER K
DOUBLE PRECISION X(20000), Y(20000), U1, U2, RK, SUMX, SUMX2, THETA, PI, R2
DOUBLE PRECISION MEANX, VARX, SDX, MEANY, VARY, SDY, RX, SEED
DOUBLE PRECISION PERCVARX, PERCSDX, PERCVARY, PERCSDY, SUMY, SUMY2
PRINT*, 'Input sample size: '
READ*, K
PRINT*, 'Input initial seed: '
READ*, SEED
PI = 3.14159265358979323846
RK = K
SUMX = 0
SUMX2 = 0
SUMY = 0
SUMY2 = 0
DO 15 I = 1, K
CALL RANDOM1(U1, RX, SEED)
SEED = RX
CALL RANDOM1(U2, RX, SEED)
SEED = RX
R2 = -2*LOG(U1)
THETA = 2*PI*U2
X(I) = SQRT(R2)*COS(THETA)
Y(I) = SQRT(R2)*SIN(THETA)
SUMX = SUMX + X(I)
SUMY = SUMY + Y(I)
SUMX2 = SUMX2 + (X(I)*X(I))
SUMY2 = SUMY2 + (Y(I)*Y(I))
CONTINUE
MEANX = SUMX/RK
MEANY = SUMY/RK
VARX = (SUMX2 - (MEANX*SUMX))/(RK-1.)
VARY = (SUMY2 - (MEANY*SUMY))/(RK-1.)
PERCVARX = 100.*(VARX - 1.)
PERCVARY = 100.*(VARY - 1.)
SDX = SQRT(VARX)
SDY = SQRT(VARY)
PERCSDX = 100.*(SDX - 1.)
PERCSDY = 100.*(SDY - 1.)
PRINT*, 'The sample mean for X is: ', MEANX
PRINT*, 'The sample mean for Y is: ', MEANY
PRINT*, 'The sample variance for X is: ', VARX
PRINT*, ' Which differs from the distribution variance by: ', PERCVARX,
'%.'
PRINT*, 'The sample variance for Y is: ', VARY
PRINT*, ' Which differs from the distribution variance by: ', PERCVARY,
'%.'
PRINT*, 'The sample std. deviation for X is: ', SDX
PRINT*, ' Which differs from the distribution std. dev. by: ', PERCSDX,
'%.'
PRINT*, 'The sample std. deviation for Y is: ', SDY
PRINT*, ' Which differs from the distribution std. dev. by: ', PERCSDY,
'%.'
END
SUBROUTINE RANDOM1(U, RX, IDUM)
DOUBLE PRECISION RX, U, IDUM
A = 16807
B = 2147483647
X = MOD(A*IDUM, B)
RX = X
U = RX/B
RETURN
END
With SEED = 5419 and sample size K = 20,000, we obtain:
Sample Mean for X = -0.00180550547.
Sample Mean for Y = -0.00462067158.
Sample Variance for X = 1.0022515, which differs from the distribution variance by 0.2252%.
Sample Variance for Y = 0.999142334, which differs from the distribution variance by -0.0858%.
Sample Standard Deviation for X = 1.00112512, which differs from the distribution standard
deviation by 0.1125%.
Sample Standard Deviation for Y = 0.999571075, which differs from the distribution standard
deviation by -0.0429%.
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