Two independent random numbers x and y from a normal

Question

Question: Two independent random numbers x and y from a normal distribution with mean 0 and standard deviation σ have joint density function p(x, y) = (1/(2πσ²)e-(x² + y²)/(2σ²) . The average z = (x + y)/2 has a one-variable probability density function of its own.

(a) Give a double integral expression for F(t), the probability that z ≤ t.

(b) Give a single integral expression for F(t). To do this, make the change of coordinates: u = (x+y)/2, v = (x - y)/2 and then do the integral on dv. Use the fact that

_∞∫^-∞ e^x2/a2dx - a√π

(c) Find the probability density function F'(t) of z.

(d) What is the name of the distribution of z?