Let X be a Bernoulli random variable Bern(θ). Let U and V be
independentnormalwithzeromeansandgivenvariancesσU2 andσV2.Assume
that X is independent of U and V . Define Z = XU +(1−X)kV where k
is a given constant. (The RV Z is a mixture of two normal random
variables and it is often used in the statistical methodology as a
Tukey noise.) Formulate for a sample from Z:
(a) The LLN.
(b) The CLT.
(Please remember that you must calculate two specific
characteristics of Z to give a complete answer.)
