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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.
 Winter '14
 gong

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