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Problems 721
(a) Verify that f(x 1 , x 2 ) is a density function.
(b) Show that x 1 and x 2 are either independent or dependent.
(c) Evaluate P(1 x 1 2, x 2 4).
(d) Find r.
★ B–42 A joint density function is
f(x 1 , x 2 ) = e K (x 1 + x 1 x 2 ), 0 … x 1 … 1, 0 … x 2 … 4
0, elsewhere
(a) Find K.
(b) Determine if x 1 and x 2 are independent.
(c) Find F x1 x 2
(0.5, 2).
(d) Find F x2ƒ x 1
(x 2 ƒ x 1 ).
B–43 Let y = x 1 + x 2 , where x 1 and x 2 are uncorrelated random variables. Show that
(a) yq = m 1 + m 2 , where m 1 = x 1 and m 2 = x 2 .
(b) s 2 where s 2 and s 2 2 = (x 2 - m 2 ) 2 1 = (x 1 - m 1 ) 2
y = s 2 1 + s 2 2 ,
.
[Hint: Use the ensemble operator notation similar to that used in the proof for Eq. (B–29).]
B–44 Let x 1 = cosu and x 2 = sin u, where θ is uniformly distributed over (0, 2p). Show that
(a) x 1 and x 2 are uncorrelated.
(b) x 1 and x 2 are not independent.
★ B–45 Two random variables x 1 and x 2 are jointly Gaussian. The joint PDF is described by Eq. (B–97),
where m 1 = m 2 = 0, s x1 = s x2 = 1 and r = 0.5. Plot f(x 1 , x 2 ) for x 1 over the range |x 1 | 5 and
x 2 = 0. Also give plots for f(x1 , x 2 ) for |x 1 | 5 and x 2 = 0.4, 0.8, 1.2, and 1.6.
B–46 Show that the marginal PDF of a bivariate Gaussian PDF is a one-dimensional Gaussian PDF.
That is, evaluate
q
f(x 1 ) = f(x 1 , x 2 ) dx 2
L
where f(x 1 , x 2 ) is given by Eq. (B–97). [Hint: Factor some terms outside the integral containing x 1
(but not x 2 ). Complete the square on the exponent of the remaining integrand so that a Gaussian
PDF form is obtained. Use the property that the integral of a properly normalized Gaussian PDF
is unity.]
B–47 The input to a receiver consists of a binary signal plus some noise. That is, assume that the input
y is y = x + n, where x is random binary data with values of ;A Volts that are equally likely to
occur. Let n be independent Gaussian random noise with a mean of m and a standard deviation
of s. Find the PDF for y as a function of A, m, and s.
B–48 Referring to your solution for Prob. B–47 and using MATLAB, plot the PDF for y, where A = 12,
m = 2, and s = 4.
B–49 Referring to your solution for Prob. B–47 and using MATLAB, calculate the probability that the
voltage at the receiver input is between 10 and 14 Volts, where A = 12, m = 2, and s = 4.
B–50 (a) y = A 1 x 1 + A 2 x 2 , where A 1 and A 2 are constants and the joint PDF of x 1 and x 2 is f x (x 1 , x 2 ).
Find a formula for the PDF of y in terms of the (joint) PDF of x.
(b) If x 1 and x 2 are independent, how can this formula be simplified?
-q