Consider the following family of distributions:P = {Pλ(X = x) : Pλ(X = x) = λxe−λ/x!; x = 0, 1, 2, . . . ; λ = 0 or 1} .This is a Poisson family with λ restricted to be 0 or 1 . Show that the family P is not complete,demonstrating that completeness can be dependent on the range of the parameter.

The Poisson distribution truncated at zero has probability function given by f (x; θ) = exp(−θ)θx A(θ)x! , x = 1, 2, ... (θ > 0), where A(θ) = 1 − exp(−θ). Let X1, . . . , Xn be a random sample of size n from this truncated Poisson distribution. (i) Show that this distribution belongs to the regular exponential distri- bution and give its canonical parameter c(θ) and the corresponding sufficient statistic T (X1, ..., Xn)

(a)The Poisson distribution truncated at zero has probability function givenbyf (x; θ) = exp(−θ)θxA(θ)x! , x = 1, 2, ... (θ > 0),whereA(θ) = 1 − exp(−θ).Let X1, . . . , Xn be a random sample of size n from this truncated Poissondistribution.(i) Show that this distribution belongs to the regular exponential distri-bution and give its canonical parameter c(θ) and the correspondingsufficient statistic T (X1, ..., Xn).

Which one is not a condition of Poisson model(a) the probability of having failures in a small time interval is constant(b) the probability of having success more than one in a small time intervalis very small(c) the probability of having success in this time interval is independent oftime ‘t’ as well as earlier success(d) the probability of having success in a small time interval (t, t+td) is Kt fora positive constant k

The Poisson distribution can model a system where two events can occur simul- taneously. a. True b. False

Which type of distribution can the Poisson model be used to approximate?Question 1Answera.Geometric distributionb.Binomial distributionc.Normal distribution