1. Let X = (X₁, X₂........Xn) be a random sample, where X_iis a Poisson random variable with meanlambda > 0.

(a) Show the distribution of T = X₁ + X₂........+ Xn is also a Poisson distribution (Hint: please use the MGF method).

(b) Show that the sample sum given by T = X₁ + X₂........+ Xn is a sufficient statistic forlambda.

(c) Show that the sample sum given by T = X₁ + X₂........+ Xn is a complete statistic forlambda .(Hint: iflambda > 0 and _x=0 (a(lamdba)_x ) = 0, then a = 0)

1. Let X = (X1, X2, ..., Xn) be a random sample, where X; is a Poisson random variable with mean 1>0. (a) (2 Points) Show the distribution of T = X1+ .. + X, is also a Poisson distribution (Hint: please use the MGF method). (b) (3 Points) Show that the sample sum given by T = X1+ .. + Xn is a sufficient statistic for (c) (2 Points) Show that the sample sum given by T = X1 + ... + Xn is a complete statistic for 1.(Hint: if 1 > 0 and _ al* = 0, then a = 0)