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Then if X = [a, b], say, solve a ≤ t−1 (y) ≤ b in terms of y. 12: the CDF Method: Suppose that the continuous cdf FX (x) is known and that Y = t(X). Find the support Y of Y . i) If t is an increasing function then, FY (y) = P (Y ≤ y) = P (t(X) ≤ y) = P (X ≤ t−1 (y)) = FX (t−1 (y)). ii) If t is a decreasing function then, FY (y) = P (Y ≤ y) = P (t(X) ≤ y) = P (X ≥ t−1 (y)) = 1 − P (X < t−1(y)) = 1 − P (X ≤ t−1 (y)) = 1 − FX (t−1 (x)). iii) The special case Y = X 2 is important. If the support of X is positive, use i).

4. , yn). ,Yik . , Yik )] exists. , yik ) yik CHAPTER 2. ,Yi k if f is a pmf. Proof. The proof for a joint pdf is given below. For a joint pmf, replace the integrals by appropriate sums. , Yik ). , yik ) dyi1 · · · dyik since the term in the brackets gives the marginal. 5. Typically E(Yi ), E(Yi2) and E(Yi Yj ) for i = j are of primary interest. Suppose that (Y1 , Y2 ) has joint pdf f(y1 , y2). Then E[h(Y1, Y2 )] = h(y1, y2 )f(y1, y2 )dy2 dy1 = χ1 χ2 h(y1, y2)f(y1 , y2 )dy1dy2 χ2 χ1 where χi are the limits of integration for dyi .

2) y2 where y1 is held fixed. 3) y1 where y2 is held fixed. 1. For n = 2, double integrals are used to find marginal pdfs (defined below) and to show that the joint pdf integrates to 1. If the CHAPTER 2. MULTIVARIATE DISTRIBUTIONS 29 region of integration Ω is bounded on top by the function y2 = φT (y1), on the bottom by the function y2 = φB (y1) and to the left and right by the lines f(y1 , y2)dy1 dy2 = f(y1 , y2)dy2 dy2 = y1 = a and y2 = b then Ω Ω b φT (y1 ) φB (y1 ) a f(y1 , y2)dy2 dy1 . Within the inner integral, treat y2 as the variable, anything else, including y1, is treated as a constant.

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