c) Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw E 2,Z(@) = X(@) +Y(@). %3D i) Show that: P(Z = z) = ) fx,x(x, z – x) ii) Now assume that X and Y are independent. Show that: P(Z = 2) = fx(x)fy(z– x) = _fx(z-y)fy(y) %3D From now on, we assume that X and Y are independent random variables which have the Poisson distributions with parameters Ax and Ay, respectively. iii) Show that Z has the Poisson distribution, with parameter Ax + Ay. iv) Show that the conditional distribution of X , given X+ Y = n , is binomial, and find its parameters.

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