UNLLI

The Unit Normal Linear Loss Integral

by

MathProf



In his classic, The Theory of Blackjack, Professsor Peter Grifin makes extensive use of the Unit Normal Linear Loss Integral (UNLLI). But what is UNLLI? The following gives a mathematical definition:

Defn of UNLLI


                          ó ¥
       UNNLI(c) =         ô    (t-c)  N(t)   dt
                     t=c  õ 

  where N(t) is the pdf for the standard Normal Distribution.

 

Since this equation may or may not appear on your browser, I will state it in words. It is the Integral, from t=c to t=infinity, of (t-c)*N(t)*dt, where N(t) is the partial distribution function of standard Normal.

There is another way to decribe this. Let n be a Normally Distributed Random variable. Then UNNLI(c) is just the expectation of the ranndom variable Lc described below:


Alternate Formula

                    
       UNNLI(c) =  Expectation of Lc      

                         ì    n-c   if   if  n>c
          for    Lc  =   í    
                         î    0   otherwise  
                 
			
          where n is distributed as a standard Normal.


In typical applications, t corresponds to a count, and Lc is measuring the gain of some play as a function of that count. It is assummed that this gain is a Linear Function, and the c represents the critical value; the gain is negative when t<c. Since the player doesn't make the play here; that is why Lc is set to 0 for this.

Note that what I am calling c is called z in Griffin on Page 91. (However, on page 38, he uses z for something else.) On page 38 of Grffin, the expression for E(n) is merely

E(n) = b * UNNLI( |m| / b )

Again, the symbol in the numerator |m| is suppossed to be the absolute value of the Greek letter mu.

To carry out the Integration for UNNLI(c), you can break the integral into two pieces: the tN(t) term and the cN(t). In the latter, c is a constant so you just end up with the Normal distribution. The former Integral can actually be evaluated in closed form. You can substitute for u=exp(-Kt^2) and du becomes -2Kt dt (K are the constants in N). This gives rise to the following additional formula:


Computational Formula

          UNLLI(z) =  pdf(z)  -  z (1 -  cdf(z) )

 where   pdf(z) = pdf for the Standard Normal Distribution   
         cdf(z) = cummulate distribution funciton for SND


UNLLI Spreadsheets

Of course, many spreadsheet programs have functions that will compute the pdf and cdf for Normally DIstributed variable. For example, in Quattro Pro ( or if you remove the @ symbols in Excel ) you could use


@NORMDIST (A22 , 0, 1, 0)- A22 * (1- @NORMDIST (A22,0,1,1) )

where A22 is the cell that contains the argument z.


As a special treat, I have provided spreadsheets with the UNLLI function in them. There are two files, UNLLI.WB2 (Quatrro Pro Format) and UNLLI.XLS (Excel Format and for IE browsers only) These have been put into the file UNLLI.ZIP, which you should be able to download using the preceeding link. You will then need to Unzip the file (with PkZip or WinZip).



May Your Earnings Always Exceed Your Expected Value!