Actuarial Outpost Applying a Riemann Sum to PDF values to get CDF
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#1
03-06-2010, 03:51 PM
 colby2152 Note Contributor SOA Join Date: Feb 2006 Location: Virginia Studying for FA, GH Core College: PSU '07 Favorite beer: Oskar Blues Old Chub Scotch Ale Posts: 4,176
Applying a Riemann Sum to PDF values to get CDF

The subject says it all... what's the validity of this, and what would be the best sum method? I am trying it out on a three-parameter Weibull distribution and the sum is leading toward values of 1.4 whereas it should be topping off at 1.

Of course, I then shrunk this Riemann sum to be equal to each point value over the maximum leading to a sum of 1, but this created a bias when I compared the expected values to observed values (generated from simulation via psuedo-random numbers looking up on these Riemann sum values).

Any thoughts from the stats and calc pros out there?
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#2
03-09-2010, 03:32 PM
 gracyjoe Member SOA Join Date: Aug 2008 Location: WI Studying for ERMMod College: UW-Whitewater Alumni Favorite beer: Samuel Adams or Spotted Cow Posts: 832

If you want to use random numbers and can run a basic program try Monte Carlo.

Create a rectangle with length equal to x - min of the domain of x, and hight at or above the max height of the distribution over then desired interval.

For example, if you want to calculate F(3) for an exponential with mean 2 then the rectangle could have verticies at (0,0) (0,3) (0,.5) (3,.5)

Now simulate n points by combining random x's over (0,3) and y's over (0,.5). Then compare the y values to the f(x). Count the number of time your points fall under the curve and divide by the total number of random points. This proportion is your cumulative probability.

50,000 or maybe even more simulations shouldn't take too long if you write a good program. If you can let your compute run for a few days, try 5 billion simulations....
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