Compute mean, variance, and moment generating function of Y

1. Suppose that we have a sample {KEZD which is constructed by K = ilei, i=1,...,n where {9;} is a set of dummy variables, i.e. D, = 1 with probabth p and D, = U with probability 1 p, and {ei} is an iid sample of Normal (n, (I2) and the:,,r are independent of each other. Here note that 5,3}, ,u, and 02 are unknown parameters. (a) lCompute mean, variance, and moment generating function of Y}. (b) Consider the linear regression of K; on the constant 1 and Di. i. Obtain the OLE estimate, salt,F 31, of the coefcient of the constant, sag,r ,31. ii. Derive the probability limit of 31 in terms of the unknown param eters. iii. Derive the asymptotic distribution of 31