Consider the set function ℙ defined for every subset of [0, 1) by the formula that ℙ (A) = 0 if A is a finite set and ℙ (A) = ∞ if A is an infinite set. Show that ℙ satisfies (1.1.3)-(1.1.5), but ℙ does not have the countable additivity property (1.1.2). We see then that the finite additivity property (1.1.5) does not imply the countable additivity property (1.1.2).
(ii) Use the formula above to show thatThis is the moment generating function for a standard normal random variable, and thus Y must be a standard normal random variable. (iii) Use the formula in (i) and Theorem 2.2.7(iv) to show that X and Y are not independent.
Consider the set function ℙ defined for every subset of [0, 1) by the formula that ℙ (A) = 0 if A is a finite set and ℙ (A) = ∞ if A is an infinite set