When shooting at a target in a twodimensional plane suppose

When shooting at a target in a two-dimensional plane, suppose that the horizontal miss distance is normally distributed with mean 0 and variance 4 and is independent of the vertical miss distance, which is also normally distributed with mean 0 and variance 4. Let D denote the distance between the point at which the shot lands and the target. Find E[D].

Solution

Let X denotes the distance of the shot lands which mean horizontal miss distance follows N(0,4)

Y denotes the distance of the target which mean vertical miss distance follows N(0,4)

D=X-Y

By using the moment generating function

D is also follows normal distribution with mean 0 and variance is 4+4=8.

E(D)=0