STAT 543 Homework
STAT 543 Homework 4 Solution
Feb. 10th, 2005
STAT 543 Homework 4 Solution
1. Problem 2.1.2 Consider n systems with failure times X 1 ,..., X n assumed to be independent and identically distributed with exponential, Σ(λ ) , distributions. (a) Find the method of moments estimate of λ based on the first moment. (b) Find the method of moments estimate of λ based on the second moment. (c) Combine your answers to (a) and (b) to get a method of moment estimate of λ based on the first two moments. (d) Find the method of moments estimate of the probability P( X 1 ≥ 1) that one system will last at least a month. Solution: Since X 1 ,..., X n i.i.d Ε(λ ) , then
∼
f ( x1 ) = λ e − λ x1 , x > 0
, and
E ( X 1 ) = ∫ xλe − λ x dx =
0 ∞
Γ(2)
λ
=
1
λ
2 …show more content…
3 ⎛1 n ⎞ ⎛1 n 3⎞ 2(⎜ ∑ x i ⎟ − ⎜ ∑ x i ⎟) ⎝n 1 ⎠ ⎝n 1 ⎠
2
4
Problem 2.2.10 Let X 1 ,..., X n denote a sample from a population with one of the following densities or frequency functions. Find the MLE of θ . (a) f ( x, θ ) = θ e −θ x , x ≥ 0;θ > 0. (Exponential density) (b) f ( x, θ ) = θ cθ x − (θ +1) , x ≥ c; c constant>0; θ > 0. (Pareto density) (c) Solution: (a).
−θ x f (x | θ ) = θ ne ∑ i = L(θ )
(θ ) = n log(θ ) − θ ∑ x i
-3-
STAT 543 Homework 4 Solution
Feb. 10th, 2005
d n ˆ ∑ xi = − ∑ xi = 0 ⇒ θ = dθ θ n n ˆ Check that ′′ = − 2 < 0 , which indicates that θ reaches the maximum of .
'
=
ˆ Hence, θ =
∑x n θ
i
is the MLE of θ . f (x | θ ) = θ nc nθ ∏ x i
−(θ +1)
(b).
(θ ) = n log θ + nθ log c − (θ + 1)∑ log x i
1
'
n
=
d n = + n log c − ∑ log x i = 0 dθ θ
ˆ ⇒θ =
Check that ′′ = −
ˆ Hence, θ = n
∑ log x
1
n
i
− n log c
n
∑ xi n θ
2
ˆ < 0 , which indicates that θ reaches the maximum of .
is the MLE of θ .
−(c +1)
(c) f (x | θ ) = c nθ nc ∏ x i ⋅ 1 [ min(x i ) ≥ θ ] , c > 0; θ >
Feb. 10th, 2005
STAT 543 Homework 4 Solution
1. Problem 2.1.2 Consider n systems with failure times X 1 ,..., X n assumed to be independent and identically distributed with exponential, Σ(λ ) , distributions. (a) Find the method of moments estimate of λ based on the first moment. (b) Find the method of moments estimate of λ based on the second moment. (c) Combine your answers to (a) and (b) to get a method of moment estimate of λ based on the first two moments. (d) Find the method of moments estimate of the probability P( X 1 ≥ 1) that one system will last at least a month. Solution: Since X 1 ,..., X n i.i.d Ε(λ ) , then
∼
f ( x1 ) = λ e − λ x1 , x > 0
, and
E ( X 1 ) = ∫ xλe − λ x dx =
0 ∞
Γ(2)
λ
=
1
λ
2 …show more content…
3 ⎛1 n ⎞ ⎛1 n 3⎞ 2(⎜ ∑ x i ⎟ − ⎜ ∑ x i ⎟) ⎝n 1 ⎠ ⎝n 1 ⎠
2
4
Problem 2.2.10 Let X 1 ,..., X n denote a sample from a population with one of the following densities or frequency functions. Find the MLE of θ . (a) f ( x, θ ) = θ e −θ x , x ≥ 0;θ > 0. (Exponential density) (b) f ( x, θ ) = θ cθ x − (θ +1) , x ≥ c; c constant>0; θ > 0. (Pareto density) (c) Solution: (a).
−θ x f (x | θ ) = θ ne ∑ i = L(θ )
(θ ) = n log(θ ) − θ ∑ x i
-3-
STAT 543 Homework 4 Solution
Feb. 10th, 2005
d n ˆ ∑ xi = − ∑ xi = 0 ⇒ θ = dθ θ n n ˆ Check that ′′ = − 2 < 0 , which indicates that θ reaches the maximum of .
'
=
ˆ Hence, θ =
∑x n θ
i
is the MLE of θ . f (x | θ ) = θ nc nθ ∏ x i
−(θ +1)
(b).
(θ ) = n log θ + nθ log c − (θ + 1)∑ log x i
1
'
n
=
d n = + n log c − ∑ log x i = 0 dθ θ
ˆ ⇒θ =
Check that ′′ = −
ˆ Hence, θ = n
∑ log x
1
n
i
− n log c
n
∑ xi n θ
2
ˆ < 0 , which indicates that θ reaches the maximum of .
is the MLE of θ .
−(c +1)
(c) f (x | θ ) = c nθ nc ∏ x i ⋅ 1 [ min(x i ) ≥ θ ] , c > 0; θ >