Unformatted text preview: 220 Chapter 4
(d) 4.55. A limiter with center—level clipping is shown in Fig. P43. (3) (b)
(d) 4.56. Let Y : 3X + 2. (a)
(d) 4.57. Find the nth moment of U, the uniform random variable in the unit interval. Repe
uniform in [(1, b].
4.58. Consider the quantizer in Example 4.20. (a) (b) Find the conditional expected value and conditional variance of X given lha One Random Variable Problems 221 Find an expression for the mean and variance of Y = g(X) for an arbitra :
uous random variable X. ,
Evaluate the mean and variance if X is a Laplacian random variable with )1
Repeat part (b) if X is from Problem 4.17 with a : 1/2. Evaluate the mean and variance if X = U 3 where U is a uniform random Va
the unit interval, [—1, l] and a = 1/2. , (c) Now suppose that when X falls in ((1, 2d), it is mapped onto the point c where d < c < 2d. Find an expression for the expected value of the mean square error:
E[(X _ c)2|d < X < 24]. (d) Find the value c that minimizes the above mean square error. 1s 0 the midpoint of
the interval? Explain why or why not by sketching possible conditional pdf shapes.
(e) Find an expression for the overall mean square error using the approach in parts c and d. Find an expression [or the mean and variance of Y = g(X) for an arbitrar
uous random variable X.
Evaluate the mean and variance if X is Laplacian with A I a 2 l and b :
Repeat part (b) if Xis from Problem 4.22, a 2 1/2, b = 3/2. Evaluate the mean and variance if X = b cos(2¢rU) where U is a uniform
variable in the unit interval [—1, '1] and a = 3/4, b : 1/2. ' ection 4.4: Important Continuous Random Variables .59. Let Xbe a uniform random variable in the interval [+2. 2]. Find and plot P| [XI > r].
In Example 4.20, let the input to the quantizer be a uniform random variable in the inter—
val [—4d, 4d]. Show that Z : X + Q(X) is uniformly distributed in l+d/2, 41/2].
Let X be an exponential random variable with parameter A. (a) For d > 0 and k a nonnegative integer, find Plikd < X < (k + l)d]. (b) Segment the positive real line into [our equiprobable disjoint intervals. The rth percentile, 77(r), of a random variable X is defined by PlX S 710)] = r/il.00. (a) Find the 90%, 95 %, and 99% percentiles of the exponential random variable with
parameter A. (b) Repeat part a for the Gaussian random variable with parameters m = 0 and 02.
Let X be a Gaussian random variable with m = 5 and (72 2 16. (a) Find P[‘X > 4],P[X 2 7],Pl6.72 < X < 10.16], P[2 < X < 7],P[6 s X S 8].
(b) PlX < a] = 0.8869, find a. (c) P[X > b] = 0.11131, find 17. (d) P[13 < X S c] = 0.0123. find c. Show that the Q-function for the Gaussian random variable satisfies Q(+x) = l — Q(x).
Use Octave to generate Tables 4.2 and 4.3. 1 Let X be a Gaussian random variable with mean m and variance (7:. (3) Find P[X S m]. (b) Find PHX ~ mi < ka].fork : 1,2,3, 4, 5, 6. (c) Find the value ofkfor which Q(k) : PtX > m + k0] : 10"]iforj = 1, 2, 3, 4, 5. 6.
A binary transmission system transmits a signal X (—1 to send a “0” bit; +1 to send a “l ” bit).The received signal is Y = X + N where noise N has a zero-mean Gaussian distrib—
utlon wrth variance 0'2. Assume that “0” bits are three times as likely as “l ” bits. (3) Find the conditional pdf of Y given the input value: fy(in 2 +1) and
frfy‘iX = “1)- .
(b) The receiver decides a “0" was transmitted if the observed value of y satisfies FIGURE P43 Find the mean and variance of Y in terms of the mean and variance of X.
Evaluate the mean and variance of Y if X is Laplacian.
Evaluate the mean and variance of Yif X is an arbitrary Gaussian randomv Evaluate the mean and variance of Y if X = b cos(27rU) where Uis a unite 7 i i i i “"1. '
domxariable 1n the unitintena “3(le z "lipiX : _1] > frfin : +1).P[X = +1} and it decides a “1” was transmitted otherwise. Use the results from part a. to show
that this decision rule is equivalent to: If y < T decide “0”; if y 2 T decide “1”. (c) What is the probability that the receiver makes an error given that a +1 was trans-
mitted? a +1 was transmitted? Assume 02 = 1/16. (d) What is the overall probability of error? Find the conditional pdf of X given that X is in the interval (d, 2d). the interval (11,261). ...
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