Problem Set 13- Jointly Gaussian Decomposition

EECS 126: Probability and Random Processes
Problem Set 13

1. Projections
The following exercises are from the note on the Hilbert space of random variables. See the
notes for some hints.
(a) Let H := {X : X is a real-valued random variable with E[X2
] < ∞}. Prove that
hX, Y i := E[XY ] makes H into a real inner product space. 1
(b) Let U be a subspace of a real inner product space V and let P be the projection map
onto U. Prove that P is a linear transformation.
(c) Suppose that U is finite-dimensional, n := dim U, with basis {vi}
n
i=1. Suppose that
the basis is orthonormal. Show that P y =
Pn
i=1hy, viivi
. (Note: If we take U = R
n
with the standard inner product, then P can be represented as a matrix in the form
P =
Pn
i=1 viv
T
i
.)
2. Exam Difficulties
The difficulty of an EECS 126 exam, Θ, is uniformly distributed on [0, 100] (i.e. continuous
distribution, not discrete), and Alice gets a score X that is uniformly distributed on [0, Θ].
Alice gets her score back and wants to estimate the difficulty of the exam.
(a) What is the MLE of Θ? What is the MAP of Θ?
(b) What is the LLSE for Θ?
3. Jointly Gaussian Decomposition
Let U and V be jointly Gaussian random variables with means µU = 1, µV = 4, respectively,
with variances σ
2
U = 2.5, σ
2
V = 2, respectively, and with covariance ρ = 1. Can we write U as
U = aV + Z, where a is a scalar and Z is independent of V ? If you think we can, find the
value of a and the distribution of Z; otherwise please explain the reason.
4. Photodetector LLSE
Consider a photodetector in an optical communications system that counts the number of
photons arriving during a certain interval. A user conveys information by switching a photon
transmitter on or off. Assume that the probability of the transmitter being on is p. If the
transmitter is on, the number of photons transmitted over the interval of interest is a Poisson
random variable Θ with mean λ, and if it is off, the number of photons transmitted is 0.
Unfortunately, regardless of whether or not the transmitter is on or off, photons may be
detected due to “shot noise”. The number N of detected shot noise photons is a Poisson
random variable N with mean µ, independent of the transmitted photons. Let T be the
number of transmitted photons and D be the number of detected photons. Find L[T | D].
1To be perfectly correct, it is possible for X = 0 but 6 E[X
2
] = 0; this occurs if X = 0 with probability 1. To fix
this, we need to define two random variables X and Y to be equal if P(X = Y ) = 1. In other words, we consider
equivalence classes of random variables, defined by the relation a.s. = . With this definition, then if X 6= 0 we do indeed
have E[X
2
] > 0.
1