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In the name of GOD.
Sharif University of Technology
Stochastic Processes CE 695 Dr. H.R. Rabiee
Homework 5 (Point Estimation)
1. Suppose x1 , x2 , ..., xn are iid samples from geometric distribution.
f (x; p) = p(1 − p)x−1
Find an estimate of p using method of moments.
2. Let (x1 , x2 , .., xn ) be iid samples from
θ
f (x|θ) = ( )|x| (1 − θ)1−|x|
2
x = −1, 0, 1
0<θ<1
(a) Find the MLE of θ.
(b) Find an estimate of θ using method of moments.
3. β is an exponential random variable with PDF fβ = λe−λβ where λ is a
constant. We observe output samples of a system x1 , x2 , ..., xn that are
x2
i.i.d and have Rayleigh PDF with parameter β.(i.e. f (x; β) = βx2 exp{− 2β
2 }).
Find a MAP estimator for β.
4. Mean Square Error of estimator W for paramter θ is
E(W − θ)2
(a) Suppose θ is a random variable with known distirbution probability
p(θ). Prove MMSE estimator of θ is W = E(θ).
(b) Consider θ is a fixed unknown variable and our estimator W is a
random variable.(Bias variance decomposition). Prove that :
Eθ (W − θ)2 = V arθ + (Eθ − θ)2 = V arθ + (Biasθ )2
(c) Why UMVUE has minimum MSE among unbiased estimators?
5. Let x1 , x2 , ..., xn be a random sample from the distribution with probability density function
f (x; θ) = θ(1 + x)−(1+θ) I(0,∞) (x)
(I denotes to the indicator function)Find the UMVUE for θ.
6. Let x1 , x2 , .., xn ∼ Bernoulli(p).
(a) Compute CR bound of p estimation.
(b) Find variance of w = x̄. Is w the UMVUE of p? why?
7. Consider x1 , x2 , ..., xn ∼ N (µ, σ 2 ).
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(a) Find MLE for σ 2 .
(b) Find UMVUE for σ 2 .
(c) Find CR bound for σ 2 . Is this equal to the variance of the estimator
of part b?
8. Let x1 , x2 , ..., xn be iid with pdf
f (x; θ) =
x
1
exp(− ), 0 ≤ x < ∞, θ > 0
θ
θ
and cdf
Z
Fθ (x) =
0
x
1
y
exp(− )dy
θ
θ
Find the UMVUE of Fθ (x). (Note: a complete and sufficient statistic for
θ is also a complete and sufficient statistic for Fθ (x)).
9. Let x1 , x2 , ..., xn be a random sample of a normal r.v. X with unknown
mean µ and variance 1. Assume that µ is itself to be a normal r.v. with
mean 0 and variance 1. Find the Bayes’ estimator of µ for the squared
loss error (MMSE Method).[Hint: you can use 4.a]
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