n this problem, we consider target detection in a radar system. Upon observing a realization x of the random variable X$$which represents the return signal in noise, a decision has to be made as to whether the target is present or absent$.$

The target amplitude is known and equal to A$>0.$ The observation is made in zero$-$mean Gaussian noise W $($real$-$valued$),$ whose variance is known and equal to $2.$ If the target is present $($hypothesis H$1),$ X$=$A$+$W$.$ I