The two probabilities are given as

where the complementary error-function (erfc(x)) is defined as

There are standard tables available for the complementary error functions.

The BER for the binary data is then given as

The BER is function of the decision threshold current. The threshold is to be optimized to get the minimum BER.

Minimum BER occurs when the threshold is

For thermal dominated regime, are equal and the threshold is mean of . That is the threshold is half way between the 0 and 1 levels.

For shot noise dominated regime, , and the threshold comes closer to .

For the optimum threshold, the BER becomes

where, Q is defined as

The quantity essentially gives the fractional noise margin of the received binary signal.

For Q > 3, the BER expression can be approximated to get

The BER is a very steep function of Q. For change in Q by 1, the BER drops almost by 3 orders of magnitude.
For a BER of better than , Q should be > 7.