It is shown that in detecting sequentially a deterministic signal 0(0 in white noise 72(0 a similar identity (iii) in theorem 2.1, to the Wald's holds concerning a stopping time r determined by making use of a likelihood ratio. It is also shown that r has finite moments of any order under quite weak conditions over the signal. The exact A. S. N. E{y} in a constant signal case has been obtained and given by (2, 8). It is also considered a detection problem of a constant signal OW a in a coloured noise based on a sub-optimal statistic which become optimal when the noise were white. Similar properties of a stopping time r to those in the white noise case have been obtained in theorem 3.1.