Download ENE621: Estimation & Detection Theory Problem Set 7 at UMD and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!
University of Maryland at College Park Department of Electrical and Computer Engineering
ENEE 621 Estimation & Detection Theory
Problem Set 7 Spring 2005
Issued: Monday, March 28, 2005 Due: Monday, April 4, 2005
Reading Assignment: H. V. Poor, Chapter IV, Section IV.C.
Problem 7.
Let y be a random variable that is uniformly distributed on [0, 1]. Suppose a random variable x is related to y via
x = f (y ) =
1 y ≤ (^12) 0 y > (^12)
We wish to develop an approximation to x of the following form
ˆx = ˆx(y ) = α + β cos(πy ) ,
where α and β are constants.
(a) Find α and β so that the mean-square approximation error E [(ˆx − x)^2 ] is mini- mized.
(b) Calculate the resulting mean-square error in your approximation.
(c) How would your answers to (a) and (b) change if y were instead uniformly dis- tributed on [0, 12 ]?
Problem 7.
Consider a communication problem whereby a scalar message is to be communicated to a receiver. From the point of view of the receiver, the message is a random variable x ∼ N (0, σ x^2 ). The transmitted signal is hx, and the receiver observes
y = hx + v ,
where v is a N (0, r) random variable. Suppose the transmitter is subject to intermittent failure, i.e., h is a random variable taking the values 0 and 1 with probability 1 − p and p, respectively. Assume h, x, and v are mutually independent.
(a) Find ˆxLLS(y ), the linear least-squares estimate of x based on observation of y , and λLLS, the associated mean-square estimation error.
(b) Prove that
E [x|y = y] =
∑^1
i=
Pr[h = i|y = y] E [x|y = y, h = i].
(c) Find ˆxBLS(y ), the Bayes least-squares estimate of x based on observation of y.
Problem 7.
Let
py (y; x) =
x if 0 ≤ y ≤ 1 /x 0 otherwise
for x > 0. Show that there exist no unbiased estimators ˆx(y ) for x. Note that because only x > 0 are possible values, an unbiased estimator need only be unbiased for x > 0 rather than for all x ∈ R.
Problem 7.
Suppose that ˆx 1 and ˆx 2 are MVU estimators of the nonrandom parameter x with λˆx 1 (x) < ∞ and λˆx 2 (x) < ∞. Show that ˆx 1 = ˆx 2. What does this say about the uniqueness of an MVU estimator? Hint: consider ˆx 3 = (ˆx 1 + ˆx 2 )/.
Problem 7.
Suppose y is a Poisson random variable with unknown mean λ > 0, i.e.,
py [k; λ] = e−λ^
λk k!
u[k] k ∈ N,
where u[k] is the discrete-time unit-step function. Show that there exists only one unbi- ased estimator of exp(− 2 λ) based on y. Comment on the usefulness of this estimator.
