FALL 2011
Instructor: Sujay Sanghavi
[email protected]
Homework 9
Due: November 22th in class
Focus: ML Estimation (Chapter 9.1 in Textbook)
Problem 1
The parameter of an exponential random variable has to be estimated from one sample. What is the ML
estimator? Is it unbiased?
Problem 2
Alice models the time that she spends each week on homework as an exponentially distributed random
variable with unknown parameter
θ
. Homework times in different weeks are independent. After spending
10, 14, 18, 8, and 20 hours in the ﬁrst 5 weeks of the semester, what is her ML estimator of
θ
?
Problem 3
X
is known to be a uniform random variable, with range
[

a, a
]
. However, the parameter
a
≥
0
is unknown,
and has to be estimated from
n
samples.
(a) What is the ML estimator?
(b) Is it unbiased?
(c) Is it consistent?
Problem 4
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
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 Spring '07
 BARD
 Normal Distribution, Standard Deviation, Variance, Probability theory, ML estimator, Sujay Sanghavi

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