Question
Suppose that X is uniformly distributed over the interval (0,1),
the pdf
i.e.
of X is
(
f (x) =
1, for 0 < x < 1;
0, otherwise
Find the pdf of Y = 2 log(X).
Answer
y
y = 2 log x x = 2e
The transformation is decreasing.
The range
ofy is 0 < y < .
Question
A certain river floods every year.
Suppose that the low water mark is set at
1 and the high water mark X has the cdf
1
F (x) = P (X x) = 1 2 , 1 x < .
x
(a) Find f (x), the pdf of X.
(b) If the low water mark is reset at zero and we use a unit of
Question
1
Let X have pdf f (x) =
(1 + x), 1 < x < 1. Find the pdf of Y =2.X
2
Answer
f (x) = 12 (1 + x), 1 < x < 1.
2
The transformation is y =
.x
It is decreasing in 1 < x 0 and increasing
in 0 < x < 1.
Also 0 < y < 1 and x =
y
dx
1
Therefore =
dy
2
Question
5
Let X have pdf f (x) = 42x
(1 x), 0 < x < 1.Find the pdf of Y = X3.
Show that the pdf integrates to 1.
Answer
The range of y is 0 < y < 1.
1
Also x = y3
dx 1 2
Therefore = y 3
dy 3
Therefore the pdf of Y is
1 2
5
1
g(y) = 42y 3 1 y3 y 3 , 0 <
Question
Suppose that X has a uniform distribution on the intervalShow
(0,1).that
1
3
the pdf of Y = (8X) is given by
(
f (x) =
3 2
y,
8
0,
Answer
The transformation is
for 0 < y < 2;
otherwise
1
1
y = (8x)3 = 2x3
Therefore the range of y is 0 < y < 2.
y3