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Main points of this exam paper are: Jointly Gaussian, Random Variables, Orthogonality, Independent, Jointly, Conditional Expectation, Uncorrelated
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Department of EECS - University of California at Berkeley EECS 126 - Probability and Random Processes - Fall 2008 Midterm 2: 11/18/
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Define “Jointly Gaussian Random Variables”
Give an example of a two orthogonal random variables that are not independent.
Give an example of two N (0, 1) random variables that are not jointly Gaussian.
Is it true that E[X|Y ] = 0 implies that X and Y are uncorrelated? Prove or provide a counterex- ample.
You flip a coin n times. The probability p that a coin toss yields H is uniformly distributed in [0, 1]. Calculate the variance of the number of Hs in the n tosses.
Let (X, Y ) be jointly Gaussian, zero mean, with var(X) = 4, var(Y ) = 1 and cov(X, Y ) = 1. Calculate E[X^2 |Y ].
Given X ∈ { 0 , 1 }, the random variable Y is exponentially distributed with rate 3X + 1 (thus, with mean (3X + 1)−^1 ).
Assume P (X = 1) = p, P (X = 0) = 1 − p. Find the MAP estimate of X given Y.
Find the MLE of X given Y.
Solve the hypothesis testing problem of X given Y with a probability of false alarm at most 10%. That is, find Xˆ as a function of Y that maximizes P [ Xˆ = 1|X = 1] subject to P [ Xˆ = 1 |X = 0] ≤ 0 .1.
For what value of p does one have the same solution for 1) and 3)?
