Random Variable - Random Signals and Processes Primer with MATLAB - page 133

If X → 0, Y → 1 , while if X → 1, Y → 0. Therefore, the output variable is

defined only in the interval [0, 1 ],

f Y ðyÞ¼ e y uðyÞ;

(2.493)

where u ( y ) is a step function

1 for y 0 ;

uðyÞ¼ (2.494)

0 otherwise :

Exercise E.2.13 The random variable X is uniform in the interval [0, 1]. Find
which monotone transformation of the variable X will result in a random variable
Y with the PDF

f Y ðyÞ¼l e ly uðyÞ: (2.495)

Answer The PDF of the variable X is:

10 x 1

f X ðxÞ¼ (2.496)

0 otherwise :

For the monotone transformation, we can write:

ð ð

y

x

f X ðxÞ d x ¼ f Y ðyÞ d y: (2.497)

0 0

Placing ( 2.495 ) and ( 2.496 ) into ( 2.497 ), we get:

ð ð

y

x

l e ly d y ¼ 1 e ly d x ¼ x ¼ : (2.498)

0 0

From ( 2.498 ), we arrive at:

1
l

y ¼ ln ð 1 xÞ (2.499)

or

1
l

Y ¼ ln ð 1 XÞ: (2.500)