Estimation of Signal and Noise Parameters from MR Data - Advance Image Processing in Magnetic Resonance Imaging - page 133

where x is a known random variable of which the PDF px

x ()

is known. Then to

find py

y ()

, it suffices to solve the equation y = g ( x ). Indeed, if the real roots of

Equation 4.167 are denoted by x , , , , …… x r such that

1

ygx = () = = g r () = , (4.168)

1

Then py

y ()

is given by

px
gx

()
()

px
gx

()
()

x 1

1

x n

py () = | ′

| ++ | ′ | +

, (4.169)

y

r

where g ′ ( x ) is the derivative of g ( x ).

4.7.1.2 Example

Let yx =

As an example, suppose the PDF px

x ()

of x is known. Then, the PDF of y ,

given by

yx = , (4.170)

is

py () = 2 ypy ( ),

2

(4.171)

y x

for y ≥ 0.

4.7.2 G ENERAL T HEOREM

Given a random vector

x = ( x n ,..., ) ,

T

(4.172)

1

whose components x are random variables and given k functions

i

g

1 ()

x ,..., g k ( , x

(4.173)

we form a new set of random variables:

y = g () x ,..., y = g ( . x (4.174)

1 k k

1

Assume k = n . To find the PDF p y ( y 1 , … , y n ) of the random vector y = ( y 1 , … , y n ) T

for a specific set of numbers y 1 , … , y n , we solve the system

g () x = ,..., y g () x = y ,

(4.175)

1 1 n n