Question

When we say Prove or disprove the following statements, “Prove” means you show the statement is true proving the correct statement using at most 3 lines or referring to a textbook theorem. “Disprove” means you show a statement is wrong by giving a counterexample why that is not true).

Are the following statements true or not? Prove or disprove these one by one. Show how the random variable X looks in each case.

(a) E[X] < 0 for some random variable X. (this means there exists some rv X such that E[X] < 0. It is possible that E[Y] > 0 for another rv Y).

(b) Var[X] < 0 for some random variable X. (this means there exists some rv X such that Var[X] < 0. It is possible that Var[Y] > 0 for another rv Y).

(c) E[X] = 1 for some random variable X.

(d) Var[X] = 1 for some random variable X.

Answer 1

(a) E(X)<0 for some random variable X. since, E(X)=sum(X*P(X)), where P(X)> =0, that is, the probability of X must be positive or minimum 0. it means random variable must be negative that is X must be negative for E(X)<0.

(b)No, the variance of any random variable cannot be negative. because if the variance is negative then the standard deviation will imaginary that's not possible. so, var(X)<0 is not possible.

(c) E(X)=1, for some random variable X, this statement is true because of E(X)=sum(X*P(X)), where P(X)> =0 and value of the random variable has whether or not positive. that means the random variable is positive or negative both possibility.

(d) var(X)=1, for some random variable X, this statement is True.