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University of Texas EE 351K Homework 10 Problem

Question # 00034793
Subject: Engineering
Due on: 12/05/2014
Posted On: 12/05/2014 12:53 AM
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A source emits a random number of photons K each time that it is triggered. We assume that the PMF of K is
pK(k;?) = c(?)e?k/?, k = 0,1,2,3...
where ? is the temperature of the source and c(?) is a normalization factor. We also assume that the photon emissions each time that the source is triggered are independent. We want to estimate the temperature of the source by triggering it repeatedly and counting the number of emitted photons.

1. Determine the normalization factor c(?).
2. Find the expected value and the variance of the number K of photons emitted if the source is triggered once.
3. Derive the ML estimator for the temperature ?, based on K1....,Kn, the numbers of photons emitted when the source is triggered n times.
4. Show that the ML estimator is consistent.
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University of Texas EE 351K Homework 10 Problem Solution

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Posted On: 12/05/2014 12:54 AM
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Preview: c(?) xxxxxxxxxxxx pK(k;?) xx the PMF xxxxxxxxx we know xxxx k=0?pKk;?=1k=0?c?*e-k?=1c?{1+e-1?+e-2?+e-3?+………………}=1c(?)*11-e-1?=1c(?)=1-e-1?Hence xxx xxxxxxxxx factor xxxx is 1-e-1? x Find the xxxxxxxx value xxx xxx variance xx the number x of photons xxxxxxx if xxx xxxxxx is xxxxxxxxx once Answer:The xxx of K xx a xxxxxxx xxxxxxxxx distribution xxxx parameter p=1-e-1?Therefore xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx variance of.....
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