EECS 126: Probability and Random Processes

Problem Set 12

1. Flipping Coins and Hypothesizing

You flip a coin until you see heads. Let

X =

(

1 if the bias of the coin is q > p.

0 if the bias of the coin is p.

Find a decision rule Xˆ(Y ) that maximizes P[Xˆ = 1 | X = 1] subject to P[Xˆ = 1 | X = 0] ≤ β

for β ∈ [0, 1]. Remember to calculate the randomization constant γ.

2. Gaussian Hypothesis Testing

Consider a hypothesis testing problem that if X = 0, you observe a sample of N (µ0, σ2

),

and if X = 1, you observe a sample of N (µ1, σ2

), where µ0, µ1 ∈ R, σ

2 > 0. Find the

Neyman-Pearson test for false alarm α ∈ (0, 1), that is, P(Xˆ = 1 | X = 0) ≤ α.

3. BSC Hypothesis Testing

Consider a BSC with some error probability ∈ [0.1, 0.5). Given n inputs and outputs (xi

, yi)

of the BSC, solve a hypothesis problem to detect that > 0.1 with a probability of false alarm

at most equal to 0.05. Assume that n is very large and use the CLT.

Hint: The null hypothesis is = 0.1. The alternate hypothesis is > 0.1, which is a composite

hypothesis (this means that under the alternate hypothesis, the probability distribution of

the observation is not completely determined; compare this to a simple hypothesis such as

= 0.3, which does completely determine the probability distribution of the observation). The

Neyman-Pearson Lemma we learned in class applies for the case of a simple null hypothesis

and a simple alternate hypothesis, so it does not directly apply here.

To fix this, fix some specific

0 > 0.1 and use the Neyman-Pearson Lemma to find the optimal

hypothesis test for the hypotheses = 0.1 vs. =

0

. Then, argue that the optimal decision

rule does not depend on the specific choice of

0

; thus, the decision rule you derive will be

simultaneously optimal for testing = 0.1 vs. =

0

for all

0 > 0.1.

4. Basic Properties of Jointly Gaussian Random Variables

Let (X1, . . . , Xn) be a collection of jointly Gaussian random variables. Their joint density is

given by (for x ∈ R

n

)

f(x) = 1

p

(2π)

ndet(C)

exp

−

1

2

(x − µ)

T C

−1

(x − µ)

,

where µ is the mean vector and C is the covariance matrix.

(a) Show that X1, . . . , Xn are independent if and only if they are pairwise uncorrelated.

1

(b) Show that any linear combination of these random variables will also be a Gaussian

random variable.

5. Independent Gaussians

Let X = (X, Y ) be a jointly Gaussian random vector with mean vector [0, 0] and covariance

matrix

2 1

1 2

Find a 2 × 2 matrix U such that UX = (X0

, Y 0

) where X0 and Y

0 are independent.

2

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