PROBABILITY
Tutorial Examples on Chagter 3 — Solutions
50 X1 ~ Bi(n, 6) so E(X1) = n6, var(X1) = n9(1 —- 6).
From information given about the conditional distribution of X2,
E(X2 | x1) = (n -— x1)¢ and -var(X2 | x1) = (n —- x1) 4) (1 — 4)
Using the laws o
PROBABILITY (LEVEL M)
Class Test December 2014
SOLUTIONS
Markers Comments
Overall, attempts at this Class Test were good.
Several students attempts at this test were hampered by not having a calculator,
and you are reminded that you should have a calculat
Tuesday, 2nd December, 2014
9.00 am 10.00 am
UNIVERSITY OF GLASGOW
School of Mathematics and Statistics
PROBABILITY/PROBABILITY (LEVEL M)
Class Test
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used in
examinations. No
PROBABILITY (LEVEL M)
Class Test December 2013
SOLUTIONS
1.
E(U )
1
(0 1 9) 4.5
10
E(U 2 )
1 2 2
(0 1 9 2 ) 28.5
10
var(U ) 28.5 (4.5) 2 8.25
2.
(i) By definition:
M X (t ) E(e Xt ) e (t ) x dx
0
This integral only converges for a negative exponent, i.e
EXAMINATION FOR THE DEGREE OF M.SC. (TAUGHT) (SCIENCE)
Probability Level M
(AMENDED)
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used in
examinations. No calculator which can store or display text or graphics may be u
Chapter 4 Sums and Means of Random Variables
4.1 Exact results for sums of random variables
We are often interested in the properties of the sum or mean of a sequence of
random variables. In this section, we will introduce methods we can sometimes use
to
Thursday, 12th December, 2013
4.30 pm 5.30 pm
UNIVERSITY OF GLASGOW
School of Mathematics and Statistics
PROBABILITY (LEVEL M)
Class Test
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used in
examinations. No calculator
PROBABILITY (LEVEL M)
Class Test December 2012
SOLUTIONS
cfw_
1.
E ( X a ) k = ( x a ) k e ( x a ) dx =
a
1
k
0
So
1
1
= Ecfw_( X a ) = E( X ) a E( X ) = a +
t k e t dt =
2
= E ( X a ) 2 = E ( X 2 ) 2 aE ( X ) + a 2
2
2
2a 2
E ( X 2 ) = 2 aE ( X ) a 2 +
Thursday, 6th December, 2012
4.30 pm 5.30 pm
UNIVERSITY OF GLASGOW
School of Mathematics and Statistics
PROBABILITY (LEVEL M)
Class Test
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used in
examinations. No calculator
4.2
The Central Limit Theorem
In the last section, we saw how to determine the
probability distribution of the sum of some sequences of
random variables. We were then able to calculate exact
probabilities associated with the sum or average. Limit
theorems
Chapter 4
Sums and Means of Random Variables
4.1 Exact results for sums of random variables
4.2 The Central Limit Theorem
4.3 The Normal approximation to discrete distributions
Probability (Level M)
Chapter 4
1
4.1 Exact results for sums of random variab
4.3 The Normal
distributions
approximation
to
discrete
The Normal distribution is commonly used to evaluate
approximate tail probabilities for the Binomial
distribution when the sample size, n, is large. The
Central Limit Theorem justifies this approximat
3.5 Covariance and correlation
1
Definition
Suppose that X is a p-dimensional random vector, and
let g(X) be any real-valued function of X. Then, if it
exists, the expected value of g(X) is defined to be:
Eg ( X )
g ( x1, x 2 , x p )p X ( x1, x 2 , x p
3.6 Functions of a random vector
1
Suppose that we wish to find the joint probability density
function of Y1, Y2, , Yp, where each Yi is a real-valued
function of the p-dimensional random vector X, i.e. Yi =
hi(X). We can write:
Y1 h1 X
Y h X
2 2
h X
Statistics 3H/5M: Introduction to R
Class test 2013 Model answers
1. (a)
(b)
(c)
(d)
R1
cats <- read.table(url( http : /www. s t a t s . gla . ac . uk/~ levers / teaching / s3i2r / cats . t x t ), header=TRUE, na.strings= *
R2
cats <- cats[complete.cases(
Statistics 3H/5M: Introduction to R
Class test: Model Answers
1. (a)
2
cushings <- read.table( c u s h i n g s . t x t , header=TRUE, na.strings= * )
(b)
4
cushings <- cushings[!is.na(cushings$Tetrahydrocortisone),]
(c)
6
cushings <- transform(cushings, T
PROBABILITY
Tutorial Examples on Chapter 4
68 Suppose that X1, X2, ., Xn are independent random variables, and that each Xi
has a Poi(ti) distribution, for some ti > 0 and > 0. Find the expected value and
variance of:
n
1 n X
(i) i ; (ii)
n i 1 t i
X
i 1
PROBABILITY
Tutorial Examgles on Chapter 3 — Solutions
39 (a) We need to find k so that the joint p.d.f. of )_( integrates to 1 over Rx.
1
££K(1+X1X2)dX2dX1 = [oi-(ix2 +—);—1X§] dx1
x2=0
1 X
= k [41+ 31de
1 1
=k[X1+—X12:|
4 0
I=§k
4
Hence, k = 4/5.
14 4 2
PROBABILITY
Tutorial Examples on Chapter 3 - continued
50 The random variable X1 has a Bi(n, ) distribution. Given that X1 = x1, the
discrete random variable X2 has a Bi(n x1, ) distribution, where 0 < < 1 .
Find E(X2) and var(X2).
51 On a certain factory
PROBABILITY
Tutorial Examples on Chapter 3
Denotes an example that is primarily intended for Masters level students.
39 The random vector (X1, X2) has the following joint probability density function (for
some real constant k > 0):
k (1 x1x 2 ),
f12 ( x1,
PROBABILITY
Tutorial Examples on Chapter 2 Continued
21
(a)
Suppose that X ~ Bi(n, ).
Use the Binomial Theorem to find Ecfw_X(X-1).
n
n 2
Hint: x( x 1) n(n 1)
x
x 2
(b)
Hence show that var(X) = n(1 ).
Hint: E(X2) = Ecfw_X(X-1) + E(X)
22
Suppose that t
PROBABILITY
Tutorial Examples for Chapter 2
13
The discrete random variable X has the following probability mass function.
x
pX(x)
(a)
-5
0.1
-4
0.2
0
0.4
2
0.2
7
0.1
(b)
(c)
Find the value of the distribution function, FX(x), at each xRX. Draw a rough
sk
PROBABILITY
Tutorial Examples for Chapter 1
1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
2
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Write down suitable sample spaces for each of the following experiments.
Decide whether each is finite, countable or uncountable. For each fini
Thursday, 30th April 2015
2.00 pm 3.30 pm
EXAMINATION FOR THE DEGREES OF M.A., M.SCI AND B.SC.
(SCIENCE)
Probability
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used in
examinations. No calculator which can store or d
Statistics 3H/5M: Introduction to R
Class test I (13 November 2014) Model answers
1. (a)
(b)
(c)
(d)
(e)
R1
cia <- read.csv( cia . csv , na.strings= ? )
R2
cia <- cia[!is.na(cia$Population),]
R3
cia$Country[which.min(cia$Population)]
R4
subset(cia, Milita
Tuesday, 29th April 2014
2.00 pm 3.30 pm
EXAMINATION FOR THE DEGREES OF M.A., M.SCI AND B.SC.
(SCIENCE)
Probability
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used in
examinations. No calculator which can store or di
3.7 The Multinomial and Multivariate
Normal distributions
The multinomial distribution and its properties
The multivariate
properties
normal
distribution
and
its
1
The multinomial distribution is the generalisation to
an arbitrary number of dimensions of