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Lecture 1. Transformation of Random Variables
Suppose we are given a random variable X with density fX(x). We apply a function g
to produce a random variable Y = g(X). We can think of X as the input to a black
box, and Y the output. We wish to find the...
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1
Lecture 1. Transformation of Random Variables
Suppose we are given a random variable X with density fX(x). We apply a function g
to produce a random variable Y = g(X). We can think of X as the input to a black
box, and Y the output. We wish to find the density or distribution function of Y . We
illustrate the technique for the example in Figure 1.1.
1
2
e-x
1/2
-1
f (x)
x-axis
X
Y
y
X-Sqrt[y]
Sqrt[y]
Y = X
2
Figure 1.1
The distribution function method finds FY directly, and then fY by differentiation.
We have FY (y) = 0 for y < 0. If y ≥ 0, then P{Y ≤ y} = P{−
√
y ≤ x ≤
√
y}.
Case 1. 0 ≤ y ≤ 1 (Figure 1.2). Then
FY (y) =
1
2
√
y +
√
y
0
1
2
e−x
dx =
1
2
√
y +
1
2
(1 − e−
√
y
).
1/2
-1 x-axis
-Sqrt[y]
Sqrt[y]
f (x)
X
Figure 1.2
Case 2. y > 1 (Figure 1.3). Then
FY (y) =
1
2
+
√
y
0
1
2
e−x
dx =
1
2
+
1
2
(1 − e−
√
y
).
The density of Y is 0 for y < 0 and
Less
