Homework 9 can be downloaded here on the ECE 302 course website.

Problem 1: Imperfect camera

A photodetector has a probability $ p $ of capturing each photon incident on it. A light source is exposed to the detector, and a million photons are captured. What is the ML estimate of the number of photons actually incident on it?

Problem 2: Imperfect Radar

A radar works by transmitting a pulse, and seeing if there is an echo. Ideally, an echo means object is present, and no echo means no object. However, some echoes might get lost, and others may be generated due to other surfaces. To improve accuracy, a radar transmits $ n $ pulses, where $ n $ is a fixed number, and sees how many echoes it gets. It then makes a decision based on this number.

Let $ p_1 $ be the probability of an echo for a single pulse when there is no object, and $ p_2 $ be the probability when there is an object. Assume $ p_1 < p_2 $. What is the max-likelihood estimation rule for whether the object is present or absent?

Problem 3: Exponential Parameter Estimation

The parameter of an exponential random variable has to be estimated from one sample. What is the ML estimator? Is it unbiased?

Problem 4: Uniform Parameter Estimation

$ X $ is known to be a uniform random variable, with range $ [-a,a] $. However, the parameter $ a \geq 0 $ is unknown, and has to be estimated from $ n $ samples. What is the ML estimator? Is it unbiased?

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