Question

# When we say Prove or disprove the following statements, “Prove” means you show the statement is...

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.

(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.

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