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722
Probability and Random Variables
Appendix B
B–51 Two independent random variables—x and y—have the PDFs f(x) = 5e -5 x u(x) and f(y) = 2e -2y u(y).
Plot the PDF for w where w = x + y.
★ B–52 Two Gaussian random variables x 1 and x 2 have a mean vector m x and a covariance matrix C x as
shown. Two new random variables y 1 and y 2 are formed by the linear transformation y = Tx.
m x = c
2
- 1 d
C 5 -2> 25
x = c d
-2> 25 4
(a) Find the mean vector for y, which is denoted by m y .
(b) Find the covariance matrix for y, which is denoted by C y .
(c) Find the correlation coefficient for y 1 and y 2 . (Hint: See Sec. 6–6.)
B–53 Three Gaussian random variables x 1 , x 2 , and x 3 have zero mean values. Three new random variables
y 1 , y 2 , and y 3 are formed by the linear transformation y = Tx, where
C x = J
6.0 2.3 1.5
2.3 6.0 2.3
1.5 2.3 6.0 K
1 1>2
T = c
1>2 1 d
T = J
5 2 -1
-1 3 1
2 -1 2 K
(a) Find the covariance matrix for y, which is denoted by C y .
(b) Write an expression for the PDF f(y 1 , y 2 , y 3 ). (Hint: See Sec. 6–6).
★ B–54 (a) Find a formula for the PDF of y = Ax 1 x 2 , where x 1 and x 2 are random variables having the
joint PDF f x (x 1 , x 2 ).
(b) If x 1 , and x 2 are independent, reduce the formula obtained in part (a) to a simpler result.
B–55 y 2 = x 1 + x 2 + x 3 , where x 1 , x 2 , and x 3 are independent random variables. Each of the x i has a onedimensional
PDF that is uniformly distributed over -(A2) x i (A2). Show that the PDF of
y 2 is given by Eq. (B–109).
★ B–56 Use the built-in random number generator of MATLAB to demonstrate the central limit theorem.
That is,
(a) Compute samples of the random variable y, where y = gx i and the x i values are obtained
from the random number generator.
(b) Plot the PDF for y by using the histogram function of MATLAB.