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evansjtriceris
Topic Author
Posts: 7
Joined: July 14th, 2002, 3:00 am

### Normal distribution > 1 for small sigma?

Has anyone run across this problem before?If the normal PDF is defined as 1/ [ sqrt(2*pi) * sigma ] * e^.... , why is the PDF mis-behaved for small values of sigma?For example - assume mu = 0 and we evaluate the PDF for x=0 (eg: e^... =1).Then, assume sigma is small - 0.01 etc. -- the PDF evaluates to 39.89 (same result occurs for the NORMDIST func. in Excel)... But, by definition, no value of the PDF can be > 1 as the integral from -inf to +inf = 1The references I've checked all say sigma must be strictly greater than zero -- but, isn't 0.3989... actually the lower limit?After a day of trying to figure this out, I'm sure there's an extremely simple answer... Thanks,Jeff.

Rez
Posts: 440
Joined: May 28th, 2003, 9:27 pm

### Normal distribution > 1 for small sigma?

There isThe CDF has to be below one not the PDF ...CDF(x) = integral[ PDF(u) du, u= -infinity to x]K.

evansjtriceris
Topic Author
Posts: 7
Joined: July 14th, 2002, 3:00 am

### Normal distribution > 1 for small sigma?

Right.. but doesn't that imply the PDF would have to take negative values somewhere for the CDF to sum to 1?Eg: if the PDF for sigma = 0.10 @ x=0 is 39.89, the PDF has to be negative somewhere else to pull the sum back below 1... but this creates negative probability events..

Posts: 25
Joined: September 5th, 2006, 10:55 am

### Normal distribution > 1 for small sigma?

No: it's an integral not a sum.As sigma tends to 0, the PDF becomes narrower and taller in the middle, but its integral over the Real domain remains at 1, and the PDF becomes a Dirac delta function.

evansjtriceris
Topic Author
Posts: 7
Joined: July 14th, 2002, 3:00 am

### Normal distribution > 1 for small sigma?

Except that doesn't work... for example... ( 1/ { sqrt(2pi) *sigma } ) * e^ ... Evaluate for mu=0, x=0, so e^.. = 1Then, as sigma -> 0, 1/ (sqrt(2pi)*sigma) tends to infinity

spv205
Posts: 478
Joined: July 14th, 2002, 3:00 am

### Normal distribution > 1 for small sigma?

as iadams said its an integral not a sumThe PDF of a discrete random variable cannot be higher than 1, since effectively your width is 1, thats not the case for a continuous random variableyou have infinite height for an infinitely small width so the area you get is 1.so the Probability that x is between +/- 1/2 for instance is still finite

Posts: 25
Joined: September 5th, 2006, 10:55 am

### Normal distribution > 1 for small sigma?

Try here. You'll find reference to the Normal function about 2/3 down the page, but read the top section explaining the concept first.The PDF is very large across a very small width, resulting in a constant integration result of 1. Remember that this is a continuous function.

croot
Posts: 104
Joined: July 23rd, 2006, 8:30 pm

### Normal distribution > 1 for small sigma?

I spy a post in the wrong section...This is the place for seasoned professionals' technobabble. Newbies, read (and learn) only!

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...

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