Download A1. The cumulative distribution function (c.d.f.) of the continuous

A1. The cumulative distribution function (c.d.f.) of the continuous random
variable X is
8
x<0
< 0
x3 0 x 1
F (x) =
:
1
1 x:
(i) Obtain the probability density (p.d.f.) function of X:
(ii) Calculate E[X 2 + 2]:
(iii) Let Y = 1
X: Derive the p.d.f. of the random variable Y: [10 marks]
1
A2. Let X and Y be discrete random variables with joint probability mass
function given by the table
1
Y
0
1
2
1
15
5
15
2
15
X
2
0
3
1
15
0
0
2
15
4
15
(i) Find the marginal distributions of X and Y:
(ii) Find the conditional distribution of Y given X = 1:
(iii) Calculate Cov(2X; 3Y ):
[10 marks]
2
A3. Let X and Y be independent random variables, each having the binomial
distribution with parameters n and p:
(i) Show that MX (t); the moment generating function (m.g.f.) of X, is
MX (t) = (pet + 1
p)n
f or
t 2 ( 1; 1):
(ii) Use MX (t) to …nd the mean and variance of X:
(iii) Stating any results to which you appeal, …nd the distribution of X + Y:
[10 marks]
3
A4. Let X and Y be continuous random variables.
(i) Show that
E[X] = E[E[X j Y ]]:
(ii) Let T be a random variable whose probability density function is
f (t) =
e
0
t
0<t<1
elsewhere:
Let X be a random variable whose probability mass function, conditional on
T = t, is
( t)k e t
k!
(a) Determine the expected value of X:
P (X = k) =
(b) Determine the variance of X:
k = 0; 1; 2; :::::::
[10 marks]
4
B5. The joint probability density function (p.d.f.) of the random variables X
and Y is given by
f (x; y) =
kx(x
0
y)
0 < x < 1;
elsewhere:
x<y<x
(i) Sketch the region for which f (x; y) is positive and show that k = 2:
(ii) Show that the marginal p.d.f.’s of X and Y are, respectively
fX (x) =
fY (y) =
8
>
<
2
3
2
3
>
: 0
4x3
0
y+
y+
0<x<1
elsewhere:
y3
3
5y 3
3
0
y<1
1<y<0
elsewhere:
Hence determine whether X and Y are independent.
(iii) Write down the conditional p.d.f. of X given Y = y and
calculate P (X > 12 j Y = 41 ):
(iv) Calculate the conditional expectation of X given Y = 14 :
5
[20 marks]
B6. The independent random variables X and Y have, respectively, probability
density functions (p.d.f.’s)
x
0<x<1
0
elsewhere;
ye y 0 < y < 1
0
elsewhere:
e
fX (x) =
fY (y) =
Let U = X=Y and V = X + Y:
(i) Show that the Jacobian of the transformation X(U; V ); Y (U; V ) is
v
(1+u)2
u
1+u
v
(1+u)2
1
1+u
:
(ii) Show that the joint p.d.f. of U and V is
fU;V (u; v) =
(
v2
v
(1+u)3 e
0
0 < u; v < 1
elsewhere:
(iii) Calculate the marginal p.d.f.’s of U and V: Determine whether U and V
are independent.
[20 marks]
6
B7.Let X be a normally distributed random variable with mean
2
and probability density function (p.d.f.)
p1
2
f (x) =
Let Y = (X
)2 =
2
)2 =2
exp[ (x
2
]
and variance
1 < x < 1:
:
(i) Show that Y has p.d.f.
g(y) =
p1
2
y
1=2
e
y=2
0
0<y<1
elsewhere:
(ii) Let X1 ; X2 ; :::; Xn be independent random variables, each having the same
distribution P
as X above.
P
Let X = n1 i Xi and, for n > 1; let S 2 = n 1 1 i (Xi X)2 . Stating any
results to which you appeal (you may assume the independence of X and S 2 ),
show that the moment generating function of (n 1)S 2 = 2 is
M (t) = (1
2t)
(n 1)=2
t < 1=2
(iii) Describe a hypothesis, test based on the statistic S 2 ; to test the null hypothesis < 0 against the alternative hypothesis > 0 : For 0 = 2; n = 10
and size 0:05; write down the exact form the test takes.
(iv) De…ne the power function of a hypothesis test. Find the smallest value of
for which the power function of the test in (iii) evaluated at is at least 0:9:
[20 marks]
7