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

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