STAT2602 Mid-term Class Test

NOT TO BE TAKEN AWAY

[Total: 50 marks]
Fall, 2024 Time: 10:30am - 12:10pm

1. Suppose that a random variable X has a p.d.f given by


3e−3x , x > 0;
f (x) =
0, otherwise.

(i) Calculate the moment generating function of X. [5 marks]

(ii) Calculate E(X) and E(X 2 ). [2 marks]
(iii) Calculate E(eX/2 ) E(eX ), and E(e4X ), if exists. [3 marks]
[Total: 10 marks]

2. The number of male and female customers who visited a company on n successive
days were recorded as (X1 , Y1 ), (X2 , Y2 ), . . . , (Xn , Yn ), such that there were Xi male
and Yi female visitors on the i-th day. Assume that for each i, Xi and Yi follow
Poisson distributions with means λ and βλ respectively, for λ, β > 0, and that the
2n observations are independent.

It is known that the Poisson distribution with mean θ > 0 has the p.d.f (or p.m.f.)
function
e−θ θx
f (x|θ) = , x = 0, 1, 2, . . .
x!
(i) Show that the likelihood function of (λ, β) based on the observations
(X1 , Y1 ), (X2 , Y2 ), . . . , (Xn , Yn ) is
Pn Pn Pn
e−n(1+β)λ β i=1 Yi λ i=1 Xi + i=1 Yi
L(λ, β) = Qn .
i=1 (Xi !Yi !)

[5 marks]
(ii) If it is known that β = 1, show that the statistic.
n
X
T = (Xi + Yi )
i=1

is sufficient for λ. [5 marks]

(iii) If both λ and β are unknown, show that the statistic
n n
!
X X
T = Xi , Yi
i=1 i=1

is sufficient for (λ, β). [5 marks]

(iv) From now on suppose that β = λ, but the value of λ is unknown.
(a) Show that the statistic
n
X
T = (Xi + 2Yi )
i=1

is sufficient for λ. [5 marks]

(b) Show that the Fisher information based on the 2n observations is
 
1
I(λ) = n 4 + .
λ

[5 marks]
(c) Show that the maximum likelihood estimator of λ is
n
r
1 T 1 X
λ̂ = + − , where T = (Xi + 2Yi )
16 2n 4 i=1

[5 marks]
(d) If n is large, the maximum likelihood estimator λ̂ has an approximately
normal distribution. State the mean and variance of this normal distri-
bution. [5 marks]
(e) It is observed that the total numbers of male and female visitors on n = 20
successive days are
20
X 20
X
Xi = 40, and Yi = 10
i=1 i=1

respectively. Calculate the maximum likelihood estimate of λ and its

standard error. [5 marks]
[Total: 40 marks]

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 A LIST OF STATISTICAL FORMULAE

dr
 
tX r
1. MX (t) = E(e ). E(X ) = MX (t) .
dtr t=0

2. For X ∼ Uniform(a, b),

x
f (x) =
b−a
3. For X ∼ N(µ, σ 2 ),

(x − µ)2
   
1 1 22
fX (x) = √ exp − , MX (t) = exp µt + σ t .
2πσ 2 2σ 2 2

n n
1X 2 1X
2
4. X = Xi . S = Xi − X .
n i=1 n i=1
h i h i2
5. Bias(θ̂) = E(θ̂) − θ. E (θ̂ − θ)2 = Var(θ̂) + Bias(θ̂) .
" 2 #  2 
∂ ln f (X; θ) ∂ ln f (X; θ) 1
6. I(θ) = E =E − . Var(θ̂) ≥ .
∂θ ∂θ2 nI(θ)
" 2 #
∂`(θ) h 2 i
7. In (θ) = E = E − ∂ ∂θ`(θ)
2 = nI(θ), where `(θ) = ln L(θ),
∂θ
L(θ) = f (X1 , X2 , . . . , Xn ; θ) for θ ∈ Ω, and f (x1 , x2 , . . . , xn ; θ) is the joint p.d.f.
for the random sample X1 , X2 , . . . , Xn .


8. Factorization: f (x1 , · · · , xn ; θ) = g T (x1 , · · · , xn ); θ h(x1 , · · · , xn ).
s
!
X
9. Expontial family: f (x; θ) = h(x)c(θ) exp pi (θ)ti (x) .
i=1

X −µ
10. CLT: √ ≈ N (0, 1) for large n.
σ/ n

θ̂ − θ
11. MLE: p ≈ N (0, 1) for large n.
1/In (θ)

********** END OF PAPER **********

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