We do need to make one assumption, estimate how large the average difficulty increase will be over the life of the rig.

So with this we can use the same formula that gives the total amount of BTC will be mined; (Bernoulli's summation of infinite geometric series formula)

http://forum.bitcoin.org/index.php?topic=7638.msg112479#msg112479

a + ar + ar

^{2}+ ar

^{3}+ ... + ar

^{k}+ ... = a/(1-r)

Eg; say your rig is currently producing 10 BTC per day, then that is around 140 BTC for the current difficulty period. This is the initial rig output, a,

a = 140 ; initial rig output of current difficulty period

Now a difficulty increase of say 50% is going to decrease your output in the next difficulty period by a third or

r = 1/1.5 = 2/3 ; 1/(difficulty multiplier)

So

**assuming**an average difficult increase of 50% for life of rig, a machine earning 140 BTC in current period can be expected to output a total of

140/(1-0.6666) = 420 BTC

Let's try something a little more optimistic, say average difficulty increases of 30%; (r = 1/1.3)

140/(1-(1/1.3)) = 607 BTC

So;

**a/(1-r)**; formula for total BTC over life of rig

where

**a**; initial rig output during current difficulty period

**r**; inverse of average difficulty multiplier (empirical r = 1/1.3)

Edit: chodpaba has done some pretty in-depth analysis of difficulty-price correlations so may have quite accurate number for average difficulty increase estimate.

Edit2: found chodpaba's data and calculated average difficulty increase over last 25 periods (50 weeks) to be 30.3%, see below.