Download M2S1 PROBABILITY AND STATISTICS II

BSc and MSci EXAMINATIONS (MATHEMATICS)
May-June 2010
This paper is also taken for the relevant examination for the Associateship.
M2S1
PROBABILITY AND STATISTICS II
Date:
Monday, 17 May, 2010
Time:
2 pm – 4 pm
Credit will be given for all questions attempted but extra credit will be given for complete or nearly
complete answers.
Calculators may not be used.
Formula sheets are included as pages 4 and 5.
c 2010 Imperial College London
M2S1
Page 1 of 5
1. (a) Explain what is meant by a σ−field A of subsets of a sample space Ω.
Let A and B belong to some σ−field A. Show that A contains the set A ∩ B.
What is meant by a probability space?
(b)
Let X and Y have joint probability density of the form
fX,Y (x, y) =
cx2 y if x2 ≤ y ≤ 1,
0
otherwise.
Find the value of c and P (X ≥ Y ).
(c)
What does it mean to say that a sequence of random variables X1 , X2 , . . . , (i) converges
in probability, (ii) converges in distribution to a random variable X?
(d)
For each of the following statements state, without proof, whether it is True or False.
(i) Maximum likelihood estimators are always unbiased.
(ii) In testing of a hypothesis H0 , the p−value of the test is the probability that H0 is
true.
(iii) Convergence in distribution always implies convergence in probability.
(iv) If a sequence of random variables X1 , X2 , . . . converges in probability to the constant
b, then E[Xn ] → b.
2. (a) Consider a random variable X with the Laplace distribution, so that the probability
density function of X is
fX (x) =
1
exp(−|x|), −∞ < x < ∞.
2
Derive the moment generating function, MX (t), of X and state the interval around 0
that t must be in for MX (t) to be well-defined.
By considering the moment generating function or otherwise, evaluate the expectation
and variance of X.
(b)
Now consider independent random variables X and Y both of which have the Laplace
distribution described in part (a).
Let U = X + Y, V = X − Y .
Derive the moment generating function of U and show that U and V are identically
distributed.
Find the joint moment generating function of U and V . Show that U and V are
uncorrelated but not independent.
M2S1 PROBABILITY AND STATISTICS II (2010)
Page 2 of 5
3. (a) Let X and Y be independent non-negative random variables, with densities fX and fY
respectively. Find the joint density of U = X and V = X + aY , where a is a positive
constant.
Let X and Y be independent and exponentially distributed random variables, each with
density
f (x) = λ exp(−λx), x ≥ 0.
Find the density of X + 12 Y . Is it the same as the density of the random variable
max{X, Y }?
(b)
Show that if Y is a random variable with moment generating function MY (t), then for
γ > 0 and t > 0,
MY (t)
P (Y ≥ γ) ≤
.
eγt
[Hint: recall Markov’s Inequality].
Hence show that if Z has a standard normal distribution, then for γ ≥ 0,
P (Z ≥ γ) ≤ e−γ
2 /2
.
4. (a) Let X1 , . . . , Xn be independent, identically distributed N (μ, σ 2 ), where both μ and σ 2
are unknown.
What is the distribution of
Pn
i=1 (Xi
− μ)2 /σ 2 ?
ˉ = n−1 Pn Xi
State, without proof, the joint distribution of the random variables X
i=1
P
ˉ 2.
and (n − 1)S 2 /σ 2 , where S 2 = (n − 1)−1 ni=1 (Xi − X)
Explain clearly how the joint distribution allows construction of an appropriate test
statistic for testing the null hypothesis H0 : μ = μ0 against the alternative hypothesis
H1 : μ 6= μ0 . Describe in detail how you would carry out the test.
(b)
Let Y1 , . . . , Yn be independent, identically distributed N (0, σ 2 ) random variables.
Pn
2
Stating, without proof,
the
distribution
of
i=1 Yi , show that the probability density
pP n
2
function of W =
i=1 Yi is of the form:
fW (w) =
(c)
2
(2σ 2 )n/2 Γ(n/2)
wn−1 exp{−w2 /(2σ 2 )}.
Let W1 , . . . , Wp be independent, identically distributed, with the probability density
fW (w) derived in part (b).
Find the maximum likelihood estimator of σ, based on W1 , . . . , Wp .
M2S1 PROBABILITY AND STATISTICS II (2010)
Page 3 of 5
M2S1 PROBABILITY AND STATISTICS II (2010)
Page 4 of 5
n∈
λ ∈ R+
θ ∈ (0, 1)
n∈
n∈
{0, 1, ..., n}
{0, 1, 2, ...}
{1, 2, ...}
{n, n + 1, ...}
{0, 1, 2, ...}
Binomial(n, θ)
P oisson(λ)
Geometric(θ)
N egBinomial(n, θ)
or
∈ (0, 1)
Z+ , θ
n+x−1 n
θ (1 − θ)x
x
x−1 n
θ (1 − θ)x−n
n−1
and the LOCATION/SCALE transformation Y = μ + σX gives
y−μ 1
y−μ
FY (y) = FX
fY (y) = fX
σ
σ
σ
MY (t) = eμt MX (σt)
0
EfY [Y ] = μ+σEfX [X]
n(1 − θ)
θ
n
θ
(1 − θ)
θ2
1 − (1 − θ)x
1
θ
(1 − θ)x−1 θ
λ
λ
e−λ λx
x!
n
θ
t
1 − e (1 − θ)
θet
1 − et (1 − θ)
θet
1 − et (1 − θ)
n
n
exp λ et − 1
1 − θ + θet
1 − θ + θet
MX
MGF
VarfY [Y ] = σ 2 VarfX [X]
n(1 − θ)
θ2
n(1 − θ)
θ2
nθ(1 − θ)
nθ
θ(1 − θ)
VarfX [X]
n x
θ (1 − θ)n−x
x
EfX [X]
θ
FX
CDF
θx (1 − θ)1−x
MASS
FUNCTION
fX
For CONTINUOUS distributions (see over), define the GAMMA FUNCTION
Z ∞
Γ(α) =
xα−1 e−x dx
∈ (0, 1)
∈ (0, 1)
Z+ , θ
Z+ , θ
{0, 1}
θ ∈ (0, 1)
PARAMETERS
Bernoulli(θ)
X
RANGE
DISCRETE DISTRIBUTIONS
M2S1 PROBABILITY AND STATISTICS II (2010)
Page 5 of 5
α, β ∈
α, β ∈ R+
R+
R+
θ, α ∈
α, β ∈ R+
R+
(0, 1)
P areto(θ, α)
Beta(α, β)
R+
ν ∈ R+
R
Student(ν)
(standard model μ = 0, σ = 1)
μ ∈ R, σ ∈ R+
R+
λ∈
R+
α<β∈R
R+
(α, β)
R
N ormal(μ, σ 2 )
(standard model β = 1)
W eibull(α, β)
(standard model β = 1)
Gamma(α, β)
(standard model λ = 1)
Exponential(λ)
(standard model α = 0, β = 1)
U nif orm(α, β)
X
PARAMS.
(x − μ)2
exp −
2σ 2
2πσ 2
1
α
Γ(α + β) α−1
x
(1 − x)β−1
Γ(α)Γ(β)
αθ α
(θ + x)α+1
− 12
ν+1
(πν) Γ
2
(ν+1)/2
ν x2
Γ
1+
ν
2
√
αβxα−1 e−βx
β α α−1 −βx
e
x
Γ(α)
λe−λx
1
β−α
fX
1−
α
θ
θ+x
1 − e−βx
1−
e−λx
FX
x−α
β−α
α
CONTINUOUS DISTRIBUTIONS
PDF
CDF
(if ν > 1)
α
α+β
θ
α−1
(if α > 1)
0
μ
Γ (1 + 1/α)
β 1/α
2
VarfX [X]
(if ν > 2)
(α +
β)2 (α
αβ
+ β + 1)
αθ2
(α − 1)(α − 2)
(if α > 2)
ν
ν−2
σ2
Γ (1 + 2/α) − Γ (1 + 1/α)2
β 2/α
α
β2
1
λ2
1
λ
α
β
(β − α)
12
(α + β)
2
EfX [X]
α
2 t2 /2}
β
β−t
e{μt+σ
λ
λ−t
MX
− eαt
t (β − α)
eβt
MGF