The Trumpbama corp. TV channel introduces a new TV format: "The wheel of life". In this format, you are first supposed to answer a trivia question. The two outcomes are obviously correct answer (X = 1), or incorrect answer (X = 0). Then you are transferred to a big wheel, which you are supposed to turn, reminiscent of roulette. However, it has only four outcomes. They show numbers 100, 200, 700, 1000. Think of this wheel as a random variable Y: if you get the outcome Y = 700, it means that you will get 700 K USD in yearly income for the rest of your life.
The idea is now that a correct answer opens up the way into a new life. The designers of the game show mean by that the following. If X = 0, the probability of getting 100 K as an outcome is 0.5, as is the probability of getting 200. The probability of getting either 700 or 1000 is zero. In contrast, if X = 1, they alter the wheel such that it produces any outcome, 100, 200, 700, or 1000, with the same probability (of obviously 0.25 or 1/4).
(a) What income do you expect, when X = 0, i.e., what is E[Y|X = 0]? (b) What income do you expect, when X = 1?
(c) Given these answers, what is E[Y|X = 1] − E[Y|X = 0], i.e.,the expected yearly income increase from answering correctly?
(d) What is the variance of income (Var(Y|X)), if X = 0? What is it, if X = 1?