Method of Moments

Introduction to the Science of Statistics
13.4 Answers to Selected Exercises
13.2. Let T be the random variable that is the angle between
the positron trajectory and the µ + -spin
µ 2 = E ↵ T 2 = 1 Z ⇡
t 2 (1 + ↵ cos t)dt = ⇡2
2⇡ ⇡
3
⇡ 2 /3)/2. This leads to the method of mo-
ˆ↵ = 1 ✓t
2
2 ⇡ 2 ◆
3
Thus, ↵ =(µ 2
ments estimate
2↵
where t 2 is the sample mean of the square of the observations.
y
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
The Method of Moments
Figure 13.2: Densities f(t|↵) for the values of ↵ = 1
(yellow). 1/3 (red), 0 (black), 1/3 (blue), 1 (light blue).
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
13.4. Let X be the random variable for the number of tagged fish. Then, X is a hypergeometric random variable with
mean µ X = kt
N and variance 2
X = k t N t N k
N N N 1
N = g(µ X )= kt . Thus, g 0 (µ X )= kt
µ X µ 2 .
X
The variance of ˆN
✓ ◆ 2
Var( ˆN) kt
⇡ g 0 (µ) 2 X 2 =
µ 2 k t ✓ ◆ 2
N t N k kt
X
N N N 1 = t kt/µ X t kt/µ X k
µ 2 k
X
kt/µ X kt/µ X kt/µ X 1
✓ ◆ 2 kt
= k µ ✓ ◆ 2
Xt kt µ X t kt kµ X kt
= k µ X k µ X k(t µ X )
kt kt kt µ X k k kt µ X
µ 2 X
= k2 t 2
µ 3 X
(k µ X )(t µ X )
kt µ X
Now if we replace µ X by its estimate r we obtain
2ˆN ⇡ k2 t 2
µ 2 X
(k r)(t r)
r 3 kt r
For t = 200,k = 400 and r = 40, we have the estimate ˆN
= 268.4. This compares to the estimate of 276.6 from
simulation.
For t = 1709,k = 6375 and r = 138, we have the estimate ˆN
= 6373.4.
.
x
203