N1281 May 2015 Exam Solution
1.
(a)
(b)
(C)
X _ 71.
[Seen] Let T 2 S/V/g' where X = 3; 21:1 XI and S =
T m tan] is a pivot for a.
Let FT he the cumulative distribution function of the 1-1 distribu
1.
Let X1 , . . . Xn be iid random variables from the pdf
x2
fX (x) = kx2 e
for 0 x < +
(1)
where > 0 is unknown and k R is the normalising constant.
(a)
(b)
3
Verify that k = 4 2 / .
Compute the ma
1.
Suppose that X and Y are independent random variables where the probability density function
of X is given by
43/2
2
fX (x) = x2 ex /
for 0 x < +
(1)
and Y Gamma (3/2, 1/). (Using the parameterizat
1.
a) State without proof whether each of the following are TRUE/FALSE:
i) Countable additivity implies finite additivity.
ii) The cumulative density function of a continuous random variable may be no
BSc and MSci EXAMINATIONS (MATHEMATICS)
May-June 2016
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M2S1
PROBABILITY AND STATISTICS II
Date:
date May 2016
Time:
time
Cre
M2S1 Autumn 2016 Solution to Mock Exam
1. (a)
Z
43/2 3 x2 /
xfX (x)dx =
E(X) =
x e
dx
0
0
r
Z
23/2 z/
23/2 2
=
ze
dz = = 2
,
0
where the third equality follows from the change of variable z = x2 and
53
cfw_0, 1, 2, .
1
(x
) ,
o
x
n
1
)x
x!
The gamma function is given by () =
0
R1
x
1
e
x
dx.
1
)
(1
n
x
x
)n
)1
1 n
(1
1
n+x
x
(1
e
n x
(1
x
x (1
fX
pmf
for x 2 RK with a (K K) variance-covarian
M251
imperial College
London
BBC, MSci and MSc EXAMINATIONS (MATHEMATICS)
May June 2015
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Pro
M2S1
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May June 2014
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Probability & Statistics II
M3S2 Spring 2015 - Problem Sheet 4 Solutions
S1) (a)
E(U ) = E
y
V ()
=0
as E(Y ) =
(b)
1
1
2 var (y ) = V ()
(V ()
var(U ) =
as
var(Y ) = V ()
(c)
E
U
=E
1
y
+ V 0 ()
V ()
(V ()2
=
1
V ()
S2) (a)
Z
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Mathematical Met
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CPRRECTEQ VEES;O~I
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EXAMINATION SOLUTIONS 2011*12 curse
' ' M x M 2.
Question
ll-
seen/unseen I EXAMINATION SOLUTIONS 201112
' MI M2.
II-
Marks 82.
. seen/unseen
My we?) F'Ere'j
x :)(%9:- 3:19.39 - t 5 613m
fkhnpyv 8'12
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IMPERIAL COLLEGE LONDON
Course:
M1A1
Setter:
Professor Andrew Parry
Checker: Dr Daniel Moore
Editor:
Dr Dmitry Turaev
External: Professor Paul Houston
Date:
April 3, 2012
BSc and MSci EXAMINATIONS (MA
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 3
1. (a) The general solution is x = a(1, 1, 1) for any a R.
(b) We spot that (1, 1, 0) is a solution to Ax = b. Thus the general
solution is (1,
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 9
1. All the claims are direct consequences of defintions.
2. (a) (i) linearly independent, do not span R3
(ii) linearly dependent since v1 + v2
M1GLA Geometry and Linear Algebra 2014
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1. Both identities follow straight from the definition of the bracket.
2. The ii-entry of AB is a1i bi1 +. . .+ani bin . The sum of these expre
M1GLA Geometry and Linear Algebra 2014
Solutions for Sheet 1
1. (i) L = cfw_(1, 2) + (3, 3)| R
(ii) n = (1, 1)
(iii) = 3
(iv) ( 34 , 53 )
2. (i) The line is cfw_a + (b a)| R, which has direction vecto
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 5
1. (a) 96; (b) solutions t = 2, 4; (c) x = 0, b c, (a + b + c)/2.
a
2. (i) Eigenvalues are 1 and 3. The corresponding eigenvectors are
a
b
1
1
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 4
1. (i) False for any pair A, B such that AB 6= BA.
0 1
(ii) False, e.g. for A = B =
.
0 0
(iii) True. Assume A1 and B 1 both exist, then
0 = A1
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Solutions Sheet 7
1. (i) The line is cfw_(3, 1, 2)+(3, 0, 6)| R. The plane is 2x1 +5x2 x3 =
9.
(ii) The plane is 2x1 x2 + x3 = 1. The line is cfw_(1, 2, 1) +
(2,