## Random Systems

### Week 3

#### Link to completed assignment

##### Problem 6.1

(a) Using Equations 6.24-6.26 from the textbook, I was able to expand and solve both sides, paying attention only to terms with k terms of the first, second, and third orders for the first three cumulants. Note that the first cumulant is the mean and the second cumulant is the variance.

(b) While the integral might look nasty, I remembered from a previous class
that the mean and variance of a Gaussian distribution are known and simply
represented by symbols *μ* and *σ ^{2}*. This
helped me simplify, but I wasn't quite able to solve completely for the
third cumulant.

##### Problem 6.2

(a) Knowing that the area of the parallelogram is equal to the base times the
height gives us a hint that after mapping *dx _{1} dx_{2}*,
the area will just be

*dy*times

_{1}*dy*.

_{2}(b)

(c)

##### Problem 6.3

(a) Using Table 6.1 from the textbook, I knew that an order 4 maximal LFSR
had *i* values of 1 and 4. Knowing this, each successive value would be
determined by the logic equation *x _{n-1} ^ x_{n-4}*. I
used "1111" as my seed value, and proceeded to fill out the bit sequence.

(b) Knowing that the number of total possible patterns is just two to the
power of the number of bits, as well as knowing that the number of patterns
divided by the clock rate in Hz gives us the total time in seconds between
repeats in the pattern helped me set up this equation. Note that while the
final answer was closer to 88 bits, you need 89 bits if you don't want to
repeat *before* the given time interval.

##### Problem 6.4

(a)

(b)

(c)

(d) Using the recommended Linear Congruential Generator (LCG) as suggested
in class, I was able to simulate 10 trajectories of a random 1D walker. I
used suggested values from Wikipedia for the best possible LCG. Those values
are *a = 1103515245, c = 12345, m = 2 ^{32}*. The plot of this
LCG's distribution, as well as the random 1D walker trajectories are below.

(e)