(monotonicity) If X ≤ Y , E[X] ≤ E[Y ]

Let X and Y be two discrete random variables with the given pmf pX (xi),i = 1, 2, . . . , n, and pY (yj ), j = 1, 2, . . . , m. Verify for following properties using thedefinition E[X] = ∑ni=1 xipX (xi).• (linearity) E[aX + bY + c] = aE[X] + bE[Y ] + c.• (monotonicity) If X ≤ Y , E[X] ≤ E[Y ].

Consider two random variables (lotteries) ˜ω1 and ˜ω2. Let ˜ω2 be dominated by ˜ω1 in the senseof the rst-degree stochastic dominance. In class we argued that the expected utility of ˜ω2 isstrictly less than of ˜ω1, i.e. E[u(˜ω2)] − E[u(˜ω1)] < 0 for any increasing utility function u(w).ˆ Explain why expected payo of ˜ω2 is also necessarily less than of ˜ω1, i.e. whyE[˜ω2] < E[˜ω1].ˆ Give an example showing that the reverse is not true. That is, give an example of tworandom variables, such that one random variable gives a better expected payo than theother, i.e. E[˜ω1] > E[˜ω2], but ˜ω2 is NOT rst-degree stochastically dominated by ˜ω1.2

if x y are two random variable then e(x,y)

Context: if x y are two random variable then e(xy) Answer question

Let X, Y be two random variables with E[Y 2] < ∞. Show that E[Y |X]is the random variable which minimizes E[(Y − f (X))2|X], and then reduce that E[Y |X]minimizes E[(Y − f (X))2], that is the best estimator for Y given X is E[Y |X]