Description

Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
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.
(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 := dimU, with basis . Suppose that the basis is orthonormal. Show that . (Note: If we take U = Rn with the standard inner product, then P can be represented as a matrix in the form

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 σU2 = 2.5, σV2 = 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
1