Let X and Y be independent Bernoulli random variables with parameter 1/2, and let Z be the random variable that returns the remainder of the division of X + Y by 2.

(a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2.

(b) Prove that X; Y;Z are pairwise independent but not mutually independent.

(c) By computing Var[X+Y +Z] according to the alternative formula for variance and using the variance of Bernoulli r.v.'s, verify that Var[X +Y +Z] =Var[X]+Var[Y ]+Var[Z]

** Kindly use the properties of Bernoulli random variables for solving questions. Thanks!