10.2.1. If x are iid from a uniform population over [0, 1], evaluate the density of x1 + + xn for (1) n = 2; (2) n = 3. What is the distribution in the general case?

10.2.2. If x1 ,... , xn are iid Poisson distributed with parameter , then (1) derive the probability function of u1 = x1 + + xn ; (2) write down the probability function of x.

10.2.3. If x1 ,... , xn are iid type-1 beta distributed with parameters (, ), then compute the density of (1) u1 = x1 + x2 ; (2) u2 = x1 + x2 + x3 . 10.2.4. Repeat Exercise

10.2.3 if the population is type-2 beta with the parameters (, ).

10.2.5. State the central limit theorem explicitly if the sample comes from (1) type-1 beta population; (2) type-2 beta population.

10.2.6. Let x n be iid Bernoulli distributed with parameter p, 0 < p < 1. Let u1 = x1 + + xn np; u2 = u1 np(1 p) ; u3 = u1 + 1 2 np(1 p) am | 08.01.19 04:37 272 | 10 Sampling distributions Using a computer, or otherwise, evaluate so that Pr{|u2 | } = 0.05 for n = 10, 20, 30, 50 and compute n0 such that for all n n0 , approximates to the corresponding standard normal value 1.96.

10.2.7. Repeat Exercise 10.2.6 with u3 of Example 10.4 and make comments about binomial approximation to a standard normal variable.

10.2.8. Let x be a gamma random variable with parameters (n, ), n = 1, 2,.... Compute the mgf of (1) u1 = x; (2) u2 = x n; (3) u3 = xn n ; (4) show that u3 goes to a standard normal variable when n .

10.2.9. Interpret (4) of Exercise 10.2.8 in terms of the central limit theorem. Which is the population and which is the sample?

10.2.10. Is there any connection between central limit theorem and infinite divisibility of real random variables? Explain. . Let x= x1++xn n . Consider the standardized x, u = x E(x) Var(x) = n j=1 (xj j ) 2 1 +

10.2.11. Let z N(0, 1). Let y = z 2 . Compute the following probabilities: (1) Pr{y u} = Pr{z 2 u} = Pr{|z| u}; (2) by using (1) derive the distribution function of y and thereby the density of y; (3) show that y 2 1 . 10.2.12. Let x1 + 2 n . Assuming the existence of the mgf of xj , j = 1,... , n work out a condition on 2 1 + + 2 n so that u z N(0, 1) as n . 10.2.13. Generalize Exercise

10.2.12 when u is the standardized v = a1 x1 + + an xn when X = (x1 ,... , xn ) has a joint distribution with covariance matrix = (ij) with < and a = (a1 ,... , an ) is a fixed vector of constants and () denotes a norm of ().