# Generating a normal random variable

1. May 17, 2006

### Physics_wiz

I'm trying to write a program in excel to generate random variables with mean mu and standard deviation sigma. I can simply refer to the worksheet function for it but it takes forever when I have it inside a loop doing a monte carlo simulation. There is one function in excel that returns a random number between 0 and 1 and I can refer to that function in the loop as many times as I want because it's not expensive (ie. doesn't take much time).

So, here's the problem:
I have mu, sigma, and a random number between 0 and 1.
I need to generate a random number from a normal distribution with mean mu and standard deviation sigma.

Anyone know how to do this?

Thanks.

2. May 17, 2006

### HallsofIvy

Staff Emeritus
I know that Knuth has an algorithm but don't remember any details.

Anyway, I googled on "Normal Distribution" and "algorithm" and got a number of hits. One you might find interesting is this:
http://www.seio.es/test/Archivos/t62/t62bun.pdf.

3. May 17, 2006

### mathman

The "standard" method (I used it years ago) is to generate normal variates in pairs.
If you use exp(-(x2+y2)/2) as the integrand, convert to polar coordinates (r,a), then let u=r2/2, you can then choose u= -ln(X), a=2pi*Y, where X and Y are random numbers, and the normal variates will be rcos(a) and rsin(a), where r=(2u)1/2.

4. May 17, 2006

### chroot

Staff Emeritus
Use the central limit theorem. If you add several uniform random variables, the result is (roughly) Gaussian.

Here's an example from a (Verilog) program:

Code (Text):

//
// Calculate roughly Gaussian noise by applying the central limit theorem
// to four [0,1] uniform random numbers. Multiply by sqrt(3) to fix the
// variance at 1.
//

noise = ( ( ($random / 4294967295.0 + 0.5) + ($random / 4294967295.0 + 0.5) +
($random / 4294967295.0 + 0.5) + ($random / 4294967295.0 + 0.5) ) - 2.0 ) * 1.732050808 * NOISE_AMPLITUDE;

The example given by mathman is the Box-Muller method, which is faster, but more difficult to understand.

http://en.wikipedia.org/wiki/Box-Muller_transform

Here's an excellent page with code for both methods:

http://www.dspguru.com/howto/tech/wgn2.htm

- Warren

5. May 17, 2006

### Physics_wiz

Ok, so I made a small program in excel to test it, but it doesn't look right. Either I'm doing something wrong or excel's random number generator isn't good enough (probably the former).

Here's the code:
Code (Text):

Sub mac()
With Sheets("Sheet1")

For counter = 1 To 1000

For counter2 = 0 To 100
Randomize
U1 = Rnd
Randomize
U2 = Rnd
V1 = 2 * U1 - 1
V2 = 2 * U2 - 1
S = V1 * V1 + V2 * V2
If S >= 1 Then counter2 = 100
Next

X = Sqr(2 * Log(S) / S) * V1
Y = Sqr(2 * Log(S) / S) * V2

.Cells(counter, 1) = X
.Cells(counter, 2) = Y
Next
End With

End Sub

This generates 2000 numbers in two columns that are supposed to be normally distrubuted with mean 0 and standard deviation 1. Right?

The problem is that the maximum number in the 2000 generated is .812241 and the minimum is -.82961. With a standard deviation of 1, I should see about 32% of the data outside of the range [-1, 1] but it looks like it's all in the range [-.8, .8]. What's going on here?

6. May 18, 2006

### Physics_wiz

I found my mistake!

Here's how the code should look:
Code (Text):

Sub mac()
With Sheets("Sheet1")

For counter = 1 To 1000

For counter2 = 0 To 100
Randomize
U1 = Rnd
Randomize
U2 = Rnd
V1 = 2 * U1 - 1
V2 = 2 * U2 - 1
S = V1 * V1 + V2 * V2
If S < 1 Then counter2 = 100
Next

X = Sqr(-2 * Log(S) / S) * V1
Y = Sqr(-2 * Log(S) / S) * V2

.Cells(counter, 1) = X
.Cells(counter, 2) = Y
Next
End With

End Sub
Now I'm getting 4 sigma events just like I should.

7. May 19, 2006

### mathman

Suggestion: try coding the algorithm I proposed. You need log, sin, cos, and sqrt functions. However, you won't have an inner loop.