tl;dr
I needed to generate Standard Normal Random Variates for a simulation I’m programming, and I wanted to see which method would be best to use. There are a few options, but generally you need to experiment to know for sure which will work best in your situation.
I decided to do this in Python, and to share my code + experimentation process and recommendations/reasoning.
Hoping this can save someone else some time later. If you need to do the same for your own simulation, or any other reason, hopefully this solves some of your problems, helps you understand what’s going on under the hood, and just makes life easier.
Grab the code here: https://github.com/adpoe/STDNormalRandVariates/tree/master/code
Generating Random Variates from Standard Normal
Experimental Setup
I used Python to algorithmically generate random variates that follow the Standard Normal distribution according to three different methods. For all methods, 10,000 valid random variables were generated in each algorithm’s run, in order to maintain consistency for later effectiveness comparisons. The methods tested were:
 The Inverse Transform Method
 The Accept/Reject Method
 The PolarCoordinates Method
In the following paragraphs, I will briefly outline the implementation decisions made to generate Standard Normal random variates according to each method. I will then analyze the results, compare them, and issue my own recommendation on which method to use going forward, informed by the data gathered in this experiment.
Inverse Transform
The Inverse Transform Method works by finding the inverse of the CDF for a given probability distribution (F^{1}(X)), then feeding random numbers generated from U[0,1) into that inverse function. This will yield randomly generated variables within the range of our desired probability distribution.
However, this method is problematic for the Standard Normal Distribution, because there is no closed form for its CDF, and hence we cannot calculate its exact inverse. Because of this, I chose to use Bowling’s closedform approximation of the Standard Normal CDF, which was developed in 2009: Pr(Z <= z) = 1 / [1 + e^(1.702z)].
Despite being only an approximation, Bowling’s closed form CDF function is mathematically close enough to generate reasonable random variates. Beyond that, this function is simple. The hardest part was calculating the inverse, which was actually done with help from Wolfram Alpha. Once an inverse was obtained, implementation was straightforward and can be seen in the code attached, within the method @inverse_transform().
Accept/Reject
The Accept/Reject method for random variate is more complex, and it can be implemented a few different ways. I chose to use the method outlined by Sheldon Ross in his book Simulation (Fifth Edition), on page 78.
The procedure, and a snippet of the core code used, are both presented inline below, as an illustration:
To note: Our Y1 and Y2 values are seeded with variates generated by the exponential distribution, with lambda=1. In much of the literature on the Accept/Reject method, this function is called “g(x)”, and for this example we used the exponential distribution.
It is also important to keep track of how many rejections we get while using the Accept/Reject method. In order to determine the average number of rejections, I ran the algorithm 10 times. The data set it created is shown below:
While generating 10,000 variates, the algorithm created 3130.8 rejections, on average. This means that, generally, there was about 1 rejected variate for every 3 valid variates.
PolarCoordinate Method
In the PolarCoordinate method, we take advantage of trigonometric properties by generating random variables uniformly distributed over (0, 2pi), and then transforming them into rectangular coordinates. This method, called the BoxMuller Method more formally, is not computationally efficient, however, because it involves use of logs, sines, and cosines—all expensive operations on a CPU.
In order to perform this operation more efficiently, I’ve used the method outlined by Sheldon Ross, on page 83 of his book Simulation (5^{th} Ed.).
Step 1: Generate random numbers, U_{1} and U_{2}
Step 2: Set V_{1} = 2U_{1} – 1, V_{2} = 2U_{2} – 1, S = V_{1}^{2} + V_{2}^{2}
Step 3: If S > 1, return to Step 1.
Step 4: Return the independent standard normal for two variables, X and Y:
Where:
X = sqrt(2*log(S)/2) * V_{1},
Y = sqrt(2*log(S)/S)*V_{2}
Expectations
Prior to running the experiment, I expected the InverseTransform Method to generate the worst variables themselves, because it only uses an approximation of the Standard Normal CDF, not the CDF itself. I was a little nervous about using an approximation for a CDF with no closed form to generate my inverse function, thinking that while our inverse may deliver results that are more or less reasonable, the resulting data set wouldn’t pass more advanced statistical tests since we are presumably losing precision, through the approximation process. But that said, I also expected its time efficiency to be the best, because we are only calculating logarithm each time we call the inverse function, and this seems to be the only slow operation.
I expected that method 2, Accept/Reject generate the most accurate variables, mostly because of the convincing mathematical proofs describing its validity on pages 77 and 78 of Ross’s Simulation textbook. Intuitively, the proof for this method makes sense, so I expected its data set to look most like something that truly follows the Standard Normal Distribution. From a time efficiency standpoint however, I expected this algorithm to perform 2^{nd} best, because I’m using a logarithm each time I generate an exponential random variable. And with 2 log calls for each run, it seems like this method would be relatively slow, under the assumption that Python’s log function is expensive. (Log calls are used here because we know that –logU is exponential with rate lambda=1. But we need exponential variables generated with rate 1 for each Y variable, Y_{1 }and Y_{2}.)
The Polar Coordinate Method is the most abstract for me, and so I had a hard time seeing exactly why it would generate Standard Normal Random variables, and because of this, I wasn’t sure what to expect of its data set. I took it on faith that it would generate the correct variables, but I didn’t fully understand why. Moreover, I also expected it to perform the worst from a runtime perspective because it involves the most expensive operations: Two Square Roots and Two Log calls for each successful run.
Histograms and Analysis
In order to determine whether each method produced qualitatively accurate data, I then plotted histograms for the numbers generated by each. All three histograms can be seen below. On my examination, it appears that the Inverse Transform yielded the least accurate representation, while the Polar Coordinate Method and Accept/Reject were much better, and about equal in validity.
Notably, the Inverse Transform method generated many values beyond the expected range of the Standard Normal (greater than 4.0 and less than 4.0). And while these values are possible, it seems like too many outliers—more than we would see in a true Standard Normal Distribution. Again, I think this is because we are using an approximation for the CDF, rather than the true Inverse of the CDF itself. I had some trouble getting these graphs to sit inline, so please review the appendix to see them.
ChiSquared Test
In order to determine wither the numbers generated may be from the Normal Distribution, I ran each method 10 times, and performed a ChiSquare test on each result. The data set can be seen in the tables within the appendix at the back of this document.
From this test, I was able to make the following determinations:
 Inverse Transform:
 N=10,000
 Avg Chi Sq: 2,806.719
 From 10 tests, each n=10,000
 Critical Value:749
 Result: REJECT Null Hypothesis
 Accept/Reject
 N=10,000
 Avg Chi Sq: 10,025.226
 From 10 tests, each n=10,000
 Critical Value: 10,233.749
 Result: ACCEPT Null Hypothesis
 Accept/Reject
 N=10,000
 Avg Chi Sq: 9,963.320
 From 10 tests, each n=10,000
 Critical Value: 10,233.749
 Result: ACCEPT Null Hypothesis
Runtime Analysis
Again, I ran each method 10 times to collect a sample of data with which to analyze runtime.
The average runtimes from 10 tests with n=10,000 were:
 Inverse Transform: 6.60E06 seconds
 Accept/Reject: 5.72E06 seconds
 Polar Coordinates: –63E06 seconds
This result was indeed surprising. I had expected the Polar Coordinates method to perform the worst, and it did—but only by a very small margin. Moreover, I had expected Inverse Transform to perform the best, and it was only fractions of a microsecond (or nanosecond?) faster than Polar Coordinates on average. I did not expected Accept/Reject to perform so well, but it was by far the fastest overall.
Conclusion
Given these results, I would recommend the Accept/Reject method for anyone who needs to generate Standard Normal Random Variables in Python (at least using my same implementations).
Accept/Reject not only passed the ChiSquare test at the 95% significance level, it also was by far the fastest of the three algorithms. It is roughly comparable to Polar Coordinates on the quality of variables generated, and it beats everything else on speed.
APPENDIX – FIGURES:
Fig 1.1 – Inverse Transform Method Histogram
Fig 1.2 – Polar Coordinates Method HistogramFig 1.3 – Accept/Reject Method Histogram
DATA ANALYSIS TABLES
INVERSE TRANSFORM 




RUN ITERATION 
CHI SQ 
CRIT VALUE, ALPHA=0.05 
NULL HYPOTHESIS 
TIME 
1 
29076.10305 
10233.7489 
REJECT 
7.87E06 
2 
28786.13727 
10233.7489 
REJECT 
5.96E06 
3 
29238.94032 
10233.7489 
REJECT 
6.20E06 
4 
27528.91629 
10233.7489 
REJECT 
8.11E06 
5 
28302.76943 
10233.7489 
REJECT 
5.96E06 
6 
28465.05791 
10233.7489 
REJECT 
5.96E06 
7 
28742.14355 
10233.7489 
REJECT 
6.91E06 
8 
29462.56461 
10233.7489 
REJECT 
5.96E06 
9 
28164.87435 
10233.7489 
REJECT 
6.20E06 
10 
28319.68265 
10233.7489 
REJECT 
6.91E06 
AVG CHI SQ: 
28608.71894 
10233.7489 
REJECT 

AVG TIME SPENT: 



6.60E06 
ACCEPT/REJECT 




RUN ITERATION 
CHI SQ 
CRIT VALUE, ALPHA=0.05 
NULL HYPOTHESIS 
TIME 
1 
9923.579322 
10233.7489 
FAIL TO REJECT 
6.91E06 
2 
10111.60494 
10233.7489 
FAIL TO REJECT 
5.01E06 
3 
9958.916425 
10233.7489 
FAIL TO REJECT 
5.01E06 
4 
10095.8972 
10233.7489 
FAIL TO REJECT 
7.15E06 
5 
10081.61377 
10233.7489 
FAIL TO REJECT 
5.96E06 
6 
10050.33609 
10233.7489 
FAIL TO REJECT 
5.01E06 
7 
9952.663806 
10233.7489 
FAIL TO REJECT 
5.01E06 
8 
10008.1 
10233.7489 
FAIL TO REJECT 
5.01E06 
9 
9953.795163 
10233.7489 
FAIL TO REJECT 
6.20E06 
10 
10115.71883 
10233.7489 
FAIL TO REJECT 
5.96E06 
AVG CHI SQ: 
10025.22255 
10233.7489 
FAIL TO REJECT 

AVG TIME SPENT: 



5.72E06 
POLAR COORDINATES 




RUN ITERATION 
CHI SQ 
CRIT VALUE, ALPHA=0.05 
NULL HYPOTHESIS 
TIME 
1 
9765.748259 
10233.7489 
FAIL TO REJECT 
5.96E06 
2 
9841.898918 
10233.7489 
FAIL TO REJECT 
4.05E06 
3 
10014.11641 
10233.7489 
FAIL TO REJECT 
5.96E06 
4 
10154.0752 
10233.7489 
FAIL TO REJECT 
7.15E06 
5 
10081.61377 
10233.7489 
FAIL TO REJECT 
7.15E06 
6 
9964.385625 
10233.7489 
FAIL TO REJECT 
5.96E06 
7 
9860.196443 
10233.7489 
FAIL TO REJECT 
4.05E06 
8 
9903.479938 
10233.7489 
FAIL TO REJECT 
1.38E05 
9 
10037.27323 
10233.7489 
FAIL TO REJECT 
7.15E06 
10 
10010.40893 
10233.7489 
FAIL TO REJECT 
5.01E06 
AVG CHI SQ: 
9963.319674 
10233.7489 
FAIL TO REJECT 

AVG TIME SPENT: 



6.63E06 

ACCEPT / REJECT – REJECTIONS 

RUN ITERATION 
NUMBER REJECTIONS 
NUMBER VARIATES GENERATED 
1 
3087 
10000 
2 
3037 
10000 
3 
3138 
10000 
4 
3129 
10000 
5 
3165 
10000 
6 
3214 
10000 
7 
3122 
10000 
8 
3198 
10000 
9 
3120 
10000 
10 
3098 
10000 
AVERAGE TOTAL REJECTIONS: 
3130.8 

AVG REJECTIONS PER VARIATE: 
0.31308 
