(a) What is the kurtosis of a normal mixture distribution that is 95 % N(0, 1) and 5 % N(0, 10)?  (b) Find a formula for the kurtosis of a normal mixture distribution that is 100p% N(0, 1) and 100(1 −...

(a) What is the kurtosis of a normal mixture distribution that is 95 % N(0, 1) and 5 % N(0, 10)?

 (b) Find a formula for the kurtosis of a normal mixture distribution that is 100p% N(0, 1) and 100(1 − p)% N(0, σ2), where p and σ are parameters. Your formula should give the kurtosis as a function of p and σ.

 (c) Show that the kurtosis of the normal mixtures in part (b) can be made arbitrarily large by choosing p and σ appropriately. Find values of p and σ so that the kurtosis is 10,000 or larger.

 (d) Let M > 0 be arbitrarily large. Show that for any p0 <>p0 and a σ, such that the normal mixture with these values of p and σ has a kurtosis at least M. This shows that there is a normal mixture arbitrarily close to a normal dis