Formalities, tips and reporting description for Introduction to Statistics 2nd project related to
Skive fjord.
In continuation of project 1 data from Skive fjord is used for a linear regression analysis in the second
project.
The assignment is formulated
1
M1S : EXERCISE SHEET 5 : SOLUTIONS
1. (i) X Binomial(500, 1/365)
fX (x) =
500
x
x
1
365
1
1
365
500x
x = 0, 1, 2, .500
(ii) If = 500/365, then
x
=
500!
1
x!(500 x)! 365
=
1
500!
1/365
x! (500 x)! 1 1/365
1
1
500x
x!
365
fX (x)
500x
1
365
x e
x!
1
x
1
x
1
M1S : EXERCISE SHEET 2 : SOLUTIONS
1.(a) Let C = event person is correctly classied. Partition according to disease status, event person is misclassied
is C = (S T ) (S T )
(b) Partition rst into correctly classied suerers/non-suerers
(i)
(i)
(D (X T )
1
M1S : EXERCISE SHEET 3
CONDITIONAL PROBABILITY AND THE PROBABILITY THEOREMS
1. Given an event F in sample space with P (F ) > 0, the conditional probability operator, dened on the
events of , satises the probability axioms (I), (II) and (III), that is,
1
M1S : EXERCISES 8
CONTINUOUS PROBABILITY DISTRIBUTIONS : PROPERTIES AND FUNCTIONALS
1. Probability models for random variables taking values on R+ may be specied via a number of probability
functions: for real valued x
Continuous CDF
FX (x) = P [X x]
PD
1
M1S : EXERCISES 9
TRANSFORMATIONS OF RANDOM VARIABLES
1. Suppose that X is a continuous random variable with range X = [0, 1], and probability density function fX
specied by
fX (x) = 2(1 x)
0x1
and zero otherwise. Find the probability distributions of r
1
M1S : EXERCISE SHEET 2
ELEMENTARY PROBABILITY
1. A person selected from a population and subjected to screening-for-a particular-disease is either a suerer (S)
or not (S ). The screening will either result in a positive test (T ), or a negative (T ). If
1
M1S : EXERCISES 6
PROPERTIES OF DISCRETE PROBABILITY DISTRIBUTIONS
1. Show that the function
1
1+
x
1+
for parameter > 0 is a valid probability mass function for a discrete random variable X taking values on
cfw_0, 1, 2, ., and nd the corresponding cumu
1
M1S : EXERCISES 7
CONTINUOUS PROBABILITY DISTRIBUTIONS
1. The length of time (in hours) that a student takes to complete a one hour exam is a continuous random
variable X with probability density function fX dened by
fX (x) = cx2 + x
0<x1
for some const
1
M1S : EXERCISE SHEET 1 : SOLUTIONS
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
AB C
ABC
ABC
ABC
(A B C ) (A B C) (A B C) (A B C)
(A B C ) (A B C ) (A B C)
(A B C ) (A B C) (A B C)
A B C
(A B C) A B C
2. (a)
(b)
(c)
(d)
(e)
TRUE
TRUE
TRUE
FALSE
TRUE
3. (a)
(b
1
M1S : EXERCISE SHEET 8 : SOLUTIONS
1. (i) By an elementary probability result
RX (x) = P [X > x] = 1 P [X x] = 1 FX (x)
(ii) By denition, and from (i),
hX (x) =
fX (x)
fX (x)
=
RX (x)
1 FX (x)
(1)
and integrating both sides with respect to x gives
HX (x
1
M1S : EXERCISE SHEET 7 : SOLUTIONS
1. (i) Density function must integrate to 1 over X = [0, 1], so
1
1
cx2 + x dx = 1 = c
fX (x) dx = 1 =
0
0
x3
x2
+
3
2
1
= 1 = c =
0
3
2
(ii) Distribution function FX given for 0 x 1 by
x
FX (x) =
fX (t) dt =
0
x3 + x2
1
M1S : EXERCISE SHEET 6 : SOLUTIONS
1. Need to show fX non-negative, and sums to one over the range of X. Sum of geometric progression gives
result, that is,
x
1
1
1
1
=
=1
fX (x) =
1+ 1+
1+
1+
x=0
x=0
Distribution function FX given, for x = 0, 1, . by
x
1
M1S : EXERCISES 4
COMBINATORICS : HYPERGEOMETRIC PROBABILITIES AND PARTITIONING
1. The binomial expansion is used to express (a + b)n in power series form:
n
(a + b)n =
k=0
n
k
ak bnk .
Use the binomial expansion to prove the following identities:
For n
1
M1S : EXERCISE SHEET 4 : SOLUTIONS
1. (i) Special case of the Binomial Expansion of (a + b)n with a = 1, b = 1.
(ii) (x + 1)m+n = (x + 1)m (x + 1)n , and
m+n
L.H.S.
: (x + 1)m+n
m+n
k
=
k=0
m
R.H.S.
: (x + 1)m (x + 1)n
=
i=0
m
m
i
n
i=0 j=0
k
=
k=0
i=0
1
M1S : EXERCISES 10
JOINT DISTRIBUTIONS AND SUMS OF RANDOM VARIABLES
1. Two fair dice are rolled. Let X and Y be the discrete random variables corresponding to the largest and
smallest of the two scores respectively. The pair (X, Y ) is a vector of rando
1
M1S : EXERCISES 5
DISCRETE PROBABILITY DISTRIBUTIONS
1. (i) The birthdays of 500 pupils in a school are recorded. Let X be the discrete random variable corresponding
to the number of these pupils who have their birthday on New Years Day.
Assuming that b
1
M1S : EXERCISE SHEET 1
EVENTS AND SAMPLE SPACES
1. Let A, B and C be three arbitrary events. Using only the operations of union, intersection and complement,
write down expressions for the following events:
(a) Only A occurs.
(c) All three events occur.
M2S1: Probability and Statistics II
Professor David A van Dyk
Statistics Section, Imperial College London
[email protected]
http:/www2.imperial.ac.uk/dvandyk
October 2015
1
CONTENTS
M2S1 Lecture Notes
Contents
1 Introduction and Motivation
1.1 Course
CHAPTER 3
DISCRETE PROBABILITY DISTRIBUTIONS
Denition 3.1.1 DISCRETE UNIFORM DISTRIBUTION
X U nif orm(n)
fX (x) =
1
,
n
x X = cfw_1, 2, ., n ,
and zero otherwise.
Denition 3.1.2 BERNOULLI DISTRIBUTION
X Bernoulli()
fX (x) = x (1 )1x ,
x X = cfw_0, 1 ,
and
2.6. TRANSFORMATIONS OF RANDOM VARIABLES
25
and, by dierentiation, because g is monotonic increasing,
fY (y) = fX (g 1 (y)
d
g 1 (t)
dt
t=y
= fX (g 1 (y)
d
g 1 (y)
dt
t=y
,
as
d
g 1 (t) > 0.
dt
Case (II): If g is decreasing, then for x X and y Y we have
g
7.3. HYPOTHESIS TESTING
65
In the Normal example given above, we have that:
z is the test statistic;
The distribution of random variable Z if H0 is true is the null distribution;
= 0.05 is the signicance level of the test (choosing = 0.01 gives a stron
2.8. JOINT PROBABILITY DISTRIBUTIONS
33
Example 2.6 We record the delay that a motorist encounters at a one-way trac stop sign. Let
X be the random variable representing the delay the motorist experiences. There is a certain
probability that there will be
6.3. MODES OF STOCHASTIC CONVERGENCE
6.3
6.3.1
57
MODES OF STOCHASTIC CONVERGENCE
CONVERGENCE IN DISTRIBUTION
Denition 6.3.1 Consider a sequence cfw_Xn , n = 1, 2, . . ., of random variables and a
corresponding sequence of cdfs, FX1 , FX2 , . so that for
CHAPTER 1
DEFINITIONS, TERMINOLOGY, NOTATION
1.1
EVENTS AND THE SAMPLE SPACE
Denition 1.1.1 An experiment is a one-o or repeatable process or procedure for which
(a) there is a well-dened set of possible outcomes
(b) the actual outcome is not known with c
1.8. COUNTING TECHNIQUES
9
NOTE : The items may be distinct (unique in the collection), or indistinct (of a unique type in
the collection, but not unique individually).
e.g. The numbered balls in the National Lottery, or individual playing cards, are dist
2.3. CONTINUOUS RANDOM VARIABLES
17
and zero otherwise. For x 1, let k(x) be the largest integer not greater than x, then
k(x)
FX (x) =
fX (xi ) =
xi x
i=1
fX (i) = 1
k(x)
1
2
and FX (x) = 0 for x < 1.
Graphs of the probability mass function (left) and c