Statistical inference - V16
Homework 3
February 08, Monday: 1215 - 1400
Realfagbygget, grupperom RFB : 4F18A 1
Version-C-160112
2016-01-13-13-04-14
Teaching assistant: Håkon Otneim2
Problem 3.1
[CB, Exercise 4.1, page 192]
A random point is distributed uniformly on
the square with vertices (1, 1), (1−1), (−1, 1),
and (−1, . − 1). That is the joint pdf is
1
2
RFB : 4F18A
Håkon Otneim
1
f (x, y) = on the square. 3 Determine the
4
probabilities of the following events,
i)
X 2 + Y 2 < 1.
ii)
2X − Y > 0.
iii) |X + Y | < 2.
f (x, y) = 4−1 1 (x, y) ∈ square , the last term is
an indicator.
3
Problem 3.2
[CB, Exercise 4.4, page 192]
A pdf is defined by 4
f (x, y) = C(x+2y) 1(0 < y < 1)1(0 < x < 2).
i)
Find the value of C.
ii)
Find the marginal distribution of X .
iii) Find the joint cdf of X and Y .
iv) Find the pdf of the random variable
Z = 9(X + 1)−2 .
4
The two terms on the right hand side are indicators, e.g. 1(0 < y < 1) = 1(0,1) (y).
Problem 3.3
[CB, Exercise 4.5a, page 192]
i)
√
Find P(X > Y ) if X and Y are jointly
distributed with pdf
f (x, y) = (x+y)1(0 ≤ x ≤ 1) 1(0 ≤ y ≤ 1)
Problem 3.4
[CB, Exercise 4.10, page 193]
The random pair (X, Y ) has distribution:
Y =2
Y =3
Y =4
X=1
X=2
X=3
1
12
1
6
1
6
1
12
1
6
0
0
1
3
0
i)
Show that X and Y are independent.
ii)
Give a probability table for random variables U and V that have the same
marginals as X and Y but are dependent.
Problem 3.5
[CB, Exercise 4.11, page 193]
Let U be the number of trials needed to get
the first head and V be the number of trials needed to get the two heads in repeated
tosses of a fair coin. Are U and V independent random variables?
Problem 3.6
[CB, Exercise 4.22, page 195]
Let (X, Y ) be a bivariate random vector with
joint pdf f (x, y)5 . Let U = aX + b and
V = cY + d , where a, b, c, and d are fixed
constants with a > 0 and c > 0. Show that
the joint pdf of (U, V ) is
fU,V (u, v) =
1
fX,Y a−1 (u − b), b−1 (v − d) .
ac
Hint: Use the transformation formula.
5
The alternative way of writing this simultaneous
density is fX,Y .
Problem 3.7
[CB, Exercise 4.26, page 195]
Let X and Y be independent random variables with X ∼ exponential(λ) 6 and Y ∼
exponential(µ). It is impossible to obtain
direct observations7 of X and Y . Instead,
we observe the random variables Z and W ,
where
Z = X∧Y = min(X, Y ) and W = 1(Z = X).
i)
Find the joint distribution of Z and
W 8.
ii)
Prove that Z and W are independent.
Hint: Show that P(Z ≤ z|W = i) =
P(Z = z) for i = 0, 1 .
6
EX = λ
This is a situation that aries, in particular, in
medical experiments. The variables X and Y are
censored.
8
cf. Stat220.
7
Problem 3.8
[CB, Exercise 4.27, page 195]
Let X ∼ N (µ, σ 2 ) and Y ∼ N (γ, σ 2 ). Suppose that X and Y are independent normal
random variables. Define U = X + Y and
V = X − Y . Show that U and V are independent normal random variables. Find the
distribution of each of them.
Problem 3.9
[CB, Exercise 4.32, page 196]
i)
For the hierarchical model
Y |Λ ∼ Poisson(Λ),
Λ ∼ gamma(α, β).
Find the marginal distribution, mean
and variance of Y . Show that the marginal
distribution of Y is negative binomial if
α is an integer.
ii)
Show that the three stage model
Y |N ∼ binomial(N, p),
N |Λ ∼ Poisson(Λ),
Λ ∼ gamma(α, β),
leads to the same marginal distribution of Y .
Textbook
George Casella and Roger L. Berger. Statistical inference. The Wadsworth &
Brooks/Cole Statistics/Probability
Series. Wadsworth & Brooks/Cole
Advanced Books & Software, Pacific Grove, CA, second edition,
2002. ISBN 0534243126.