Homework 1 (Due: February 2)

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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 1
Due: February 2, 2017

Reading
Madhow: Appendix A and Section 3.1.

Note: the material in sections A.1 and A.2 has been covered in ECE 528 and you are expected to be familiar and comfortable with that material.

Problems
These problems are review problems for probability and random variables.
  1. A noisy discrete communication channel is available. Once each second one letter from the three-letter alphabet {a,b,c} can be transmitted and one letter from the three-letter alphabet {1, 2, 3} is received. The conditional probabilities of the various received letters, given the various transmitted letters are specified by the diagram in the accompanying diagram.

    PIC

    The source sends a, b, and c with the following probabilities:

    P [a] =   0.3
 P[b] =   0.5

 P[c] =   0.2
    1. Compute all (nine) conditional probabilities of the form P(X|Y ) for X ∈{a,b,c} and Y ∈{1, 2, 3}.
    2. Compute all (nine) joint probabilities of the form P(X,Y ) for X ∈{a,b,c} and Y ∈{1, 2, 3}.
    3. A receiver makes decisions as follows:
      • If 1 is received, decide a was sent.
      • If 2 is received, decide b was sent.
      • If 3 is received, decide c was sent.

      What is the probability that this receiver makes a wrong decision? (I.e.., its decision is different from what was actually sent.)

    4. What is the best receiver decision rule (assignment from 1, 2, 3 to a, b, c)?
    5. What is the resulting probability of error?
  2. Consider a random variable X having a double-exponential (Laplacian) density,
    p  (x ) = ae-b|x|,- ∞ <  x < ∞
 X

    where a and b are positive constants.

    1. Determine the relationship between a and b such that pX(x) is a valid density function.
    2. Determine the corresponding probability distribution function PX(x).
    3. Find the probability that the random variable lies between 2 and 3.
    4. What is the probability that X lies between 2 and 3 given that the magnitude of X is less than 3.
  3. Let x1, x2, , xN be a set of N identically distributed statistically independent random variables, each with density function px and distribution function Fx. These variables are applied to a system that selects as its output, yN, the largest of the {xi}, i.e., yN = max{x1,x2,,xN}. Clearly, yN is a random variable.
    1. Express pyN in terms of N, px, and Fx.
    2. Assume now that the xi are exponentially distributed random variables:
              {   -α
p (α) =    e    α ≥  0,
 x          0   α <  0.

      Calculate the expectation E[yN] for N = 1, 2.

  4. Let ⃗X be a zero mean Gaussian random vector with covariance matrix K.
         ⌊             ⌋
         3   - 3 0
K  = ⌈  - 3  5   0 ⌉
         0   0   8

    1. Give an expression for the density function fX⃗(x).
    2. If Y = X1 + 2X2 - X3, find fY (y).
    3. If the vector ⃗
Z has components defined by
      Z1   =  5X1  - 3X2 -  X3
Z    =  - X  +  3X  - X
  2         1      2    3
Z3   =  X1  + X3

      determine f⃗Z(⃗z). What are the properties of the new random vector?

    4. Determine fX1|X2(x1|x2 = β)