Statistics: distributions

1. Suppose the random variables X and Y are iid and distributed as N(0,1).
Let U=X+Y, V=X-Y

a) Determine the marginal pdf of U and V
b) Compute the probability P (U<0, V>0)

2. Suppose X1,....,Xn is a random sample drawn from the geometric distribution G(p). Determine the parameter space for the geometric model

3. Show that if W~ F(k1,k2), i.e. W has the F-distribution with parameter k1 and k2, then 1/W ~ F(k2,k1)

4. Show that if T ~ t(k), ie. T has the t distribution with parameter k, then T^2 ~ F(1,k)

Please explain in details. Thank you so much1