a. If Xi,.., Xn are independent Poisson distributed with parameters (Xi) respectively, then from problem 2.14c, we have
MXi (t) = eAi(e-1) for i = 1,2,… ,n
Y = Xi has My(t) = П MXi (t) since the X/s are independent. Hence
MY(t) = ei=1
which we recognize as a Poisson with parameter ^ Xi.
b. IfXi, ..,Xn are IIN (^i, a2), then from problem 2.14b, we have MXi(t) = ew‘+ 1ai2‘2 for i = 1,2,.., n
Y = ^ Xi has MY(t) = ]""[ MXi (t) since the X/s are independent. Hence
MY(t) = e 1=i
c. If Xi,.., Xn are IIN(|a,, a2),thenY = J2 Xi is N(np,,na2) from part b using
the equality of means and variances. Therefore, X = Y/nis N(p,, a2/n).
d. If Xi,.., Xn are independent x2 distributed with parameters (ri) respectively, then from problem 2...Read More