ex 2 2 2 0 1 1 course notes were prepared by dr rmap

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Unformatted text preview: e dy fX (x) = 4 −∞ −∞ 4 ∞ 1 1 1 ∞ e−y dy = e−|x| × −e−y 0 = e−|x| . = e−|x| 2 2 2 0 ∞ =1 1 Course notes were prepared by Dr. R.M.A.P. Rajatheva and revised by Dr. Poompat Saengudomlert. 1 Suppose that we want to evaluate Pr{X ≤ 1, Y ≤ 0}. It can be done as follows. 0 Pr{X ≤ 1, Y ≤ 0} = 1 −∞ −∞ 1 1 −|x|−|y| e dxdy 4 1 e−|x| dx 4 −∞ 1 x0 e |−∞ − e−x = 4 0 e−|y| dy = −∞ 1 0 ey |0 −∞ = 2− 1 e =1 =2−e−1 2.2 1 4 Functions of Random Variables Consider a random variable Y that is obtained as a function of another random variable X . In particular, suppose that Y = g (X ). We first consider g that is monotonic (either increasing or decreasing). Monotonic Functions If g is monotonic, each value y of Y has a unique inverse denoted by g −1 (y ), as illustrated in figure 2.3. Figure 2.3: Monotonic function of random variable X . When g is monotonically increasing, g − 1 (y ) −1 FY (y ) = Pr{Y ≤ y } = Pr{X ≤ g...
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This note was uploaded on 01/28/2014 for the course ELECTRONIC TC401 taught by Professor Gong during the Winter '14 term at Aarhus Universitet.

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