Response to the following problems:
(i) Let {Xi} be a sequence of rv's that converges in probability to a constant a. Assume that P(Xi > 0) = 1.
a) Show that the sequence {Yi} defined by Yi = sqrt(Xi) converges in probability to sqrt(a).
b) Show that, if a>0, the sequence {Zi} defined by Zi = a/Xi converges in probability to one.
(ii) Given that N = n the conditional distribution of Y is X_2n^w. The unconditional distribution of N is Poisson (theta).
a) Calculate E(Y) and Var(Y), i.e., the unconditional moments.
b) Show that, as theta -> infinity
[Y - E(Y)]/[sqrt(var(Y)] ---d--> N(0,1)
(iii) Let X be a rv with a Student's t distribution with p degrees of freedom.
a) Calculate E(X) and Var(X).
b) Show that X^2 has an F1,p distribution, ie. an F distribution with 1 and p degrees of freedom.
c) Use the results of parts (a) and (b) to argue that, as p --> infinity, X^2 converges in distribution.
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