A certain segment of the sky contains Z stars. Each star is either a red giant with probability 1/3 or a white dwarf with probability 2/3, independently from the other stars.
(a) If we condition on the event {Z = 4}, what is the joint p.m.f. of the red giants X and the white dwarves Y ? Fill in a 5 × 5 table with the (f (x, y))4x,y=0 values.
(b) Show that if Z ? POI(6), then the random variables X and Y are independent and have Poisson distribution by calculating the joint p.m.f. of X and Y.