Question

5. Suppose X ~ Poisson(A = 5) and Y ~ Poisson(λ = 10), and they are independent. Using the moment generating function method, find the distribution of Z = X + Y.

Answer 1

here as we know that mgf of poisson distribution: M(t)=$e^{\lambda (e^{t}-1) }$

therefore mgf of X: M_x(t)=$e^{\5 (e^{t}-1) }$

mgf of Y: M_x(t)=$e^{\10(e^{t}-1) }$

therefore mgf of Z =M_z(t)=M_X+Y(t)=M_X(t)*M_Y(t) =$e^{\5 (e^{t}-1) }$*$e^{\10(e^{t}-1) }$ =$e^{\15(e^{t}-1) }$

for mgf of Z follows Poisson distribution mgf with parameter $\lambda$ =15

hence distribution of Z is Poisson with parameter $\lambda$ =15