2b)
It is a linear model because as seen from the scatter plot, data plots of volume and Y seem to fall along a straight line.
xi y i
xi
n
i=1
yi
n
i=1
x i2
n
i=1
xi
n
i=1
n
n
i=1
n
^b=
28 (368.974 ) ( 465.04)(20.16)
28 ( 8189.164 )( 465.04)2
956.0656
130
Question 1a
X Bernoulli ( p )
Let X be the number of orders with a higher price
p is the proportion of orders with higher price
H 0 : p=0.5
H 1 : p>0.5
Question 1b
Since we are investigating on the proportion and hence, probability of price
increasing and
Statistical Estimation
Hypothesis Testing
Interval Estimation
Confidence interval
- Level of confidence: a measure of our level of belief
- Margin of error: a measure of the precision of our
estimate
Quality of regression
(Residual Plots)
Correlation & Re
Probability
Probability: is the long run frequency
of an outcome.
Independent
P(A and B)=P(A)P(B)
A & B are independent means the
occurrence of one event d oes n ot
change the chance of the other.
Disjoint/Mutually exclus
STAT203 Financial Mathematics
Assignment 2, Term 2, 2016 2017
To be handed in by: 17 January (Tues), 7:00 pm
1. You are given the following:
2t3 + 8t
, 0 t 1,
t4 + 8t2 + 16
(b) i = equivalent annual effective rate over the first year, given (t) in (a),
(a
STAT203 Financial Mathematics
Assignment 3, Term 2, 2016 2017
To be handed in by: 24 January (Tues), 7:00 pm
1. Vicky wants to purchase a perpetuity paying 100 per year with the first payment
due at the end of year 11. She can purchase it by either
(a) pa
STAT203 Financial Mathematics
Assignment 4, Term 1, 2016 2017
To be handed in by: 16 September (Friday), 3:30 pm
1. An insurance company has an obligation to pay the medical costs for a claimant.
Average annual claims costs today are $5000, and medical in
STAT203 Financial Mathematics
Assignment 1, Term 2, 2016 2017
To be handed in by: 10 January (Tues), 7:00 pm
1. (a) Given r(2) = 5%, find the effective rate i.
(b) Given an effective rate of i = 5.26%, find r(6) with the same i.
(c) Given r(4) = 0.07, fin
STAT203 Financial Mathematics
Assignment 5, Term 2, 2016 2017
To be handed in by: 7 February (Tuesday), 7:00 pm
1. John buys an annuity at a price X that will give him a yield rate of 5% effective.
The annuity consists of 10 payments at 3-year intervals,
3a)
f ( s )=
For
1
k s k1 es , k =2
( k 1 ) !
1
2 s 21 e s
( 21 ) !
2
s
se
=1,
3
P ( s> 3 )=1P ( s 3 )
1 ( 12 s es ) ds
10.8009
0.1991
0
For
=2,
3
P ( s> 3 )=1P ( s 3 )
1 ( 22 s e2 s ) ds
10.9826
0.01735
0
Therefore, since P ( s> 3|=1 )> P ( s>3|
1ai)
X is the number of pieces of carrot
X~Bin (20,0.1)
Y is the number of pieces of hay pellet
Y~Bin (20,0.5)
E(A)
= P(X10)(1) + P(X<10) * (1+3)
= [1 - P(X9)] + P(X9) * 4
= $3.9999785
$4
E(B)
= P(Y10)(3) + P(Y<10) * (1+3)
= [1 - P(X9)] * 3 + P(X9) * 4
=
1a)
He should choose an exponential distribution as he is looking for the time to the
recovery of the bonds.
He should not choose a Poisson distribution because he is not looking for the
number of occurrences of the recovery of bonds within a period.
He s
4a)
P(Sn|Sn-1) =
P( S
|Sn-1) =
n
4b)
P( S
S2
1
= P(S2
= P( S
S
3
S
S4 | S
S4 | S
3
= P(S4 | S
S4)
3
S
) P( S
3
4c)
p ( S n )= p ( Sn Sn1 ) + P ( S n S n1 )
P ( Sn S n1 )
P ( Sn1 )
P ( Sn1 ) +
P ( S n Sn1 )
P ( S n1 )
P ( S n1 )
P ( S n|S n1 ) P (
Q2a)
H0: P (Bonds recover) 0.1
H1: P (Bonds recover) 0.1
( )=12
( ) 12
=
=
=0.1
n
120
Y ~ Bin (120, 0.1)
12012
12
(Y =12 )= 120 (0.1) (10.1 )
12
( )
0.12054> 0.05
Therefore, H0 is not rejected.
By Central Limit Theorem, we approximate P to a Normal Di
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 9
(1) An insurance company issues 5000 medical insurance policies. The number of
claims filed by a policy-holder under the insurance policy during one year has the
following probability
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 8 Solutions
(1) (Continuation of Classwork 5)
(a)
fX (x) =
Z
fX (x) =
E(X) =
0
2
x
Z
xy
x y 2 2
x3
dy = ( ) = x
2
2 2 x
4
2
x3
4
x
for 0 < x < 2
0
elsewhere
x3
x3 x5 2
= 1.0667
x (x )d
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 8
(1) (Continuation of Classwork 5)
probability density function
Let X and Y be continuous random variables with joint
f(x, y) =
xy/2
for 0 < x < y < 2
0
elsewhere
(a) Find E(X), E(Y ) a
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 11
(1) You are trying to set up a portfolio that consists of Fund X and Fund Y only. The following
information provide the joint probability distribution of X and Y in terms of the annua
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 9 Solutions
(1)
(a) Let X be the number of claims by a policyholder during one-year period.
E(X) = 0 0.75 + 1 0.12 + 2 0.08 + 3 0.05 = 0.43
E(X 2 ) = 02 0.75 + 12 0.12 + 22 0.08 + 32 0.0
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 7
(1) Given that a random sample of size 5 with y1 = 1.3, y2 = 2.4, y3 = 0.9, y4 = 2.1, y5 =
1.6 is drawn from the probability model
fY (y) =
1
22
for 1 2 < y < 1 + 2
0
elsewhere
(i) Fin
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 10 Solutions
(1) Let Yi be the random variable that policy-holder i makes a claim. Yi = 1 if he makes a
claim and Yi = 0 if he does not make a claim.
Let Xi be the random variable that t
Term 1, 16/17
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Classwork 10
(1) An insurance company provides insurance to 800 independent policy-holders with following characteristics:
For Each Insured and Given a Claim
Number
Probability of Claim
Expected
V
Term 1, 15/16
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Assignment 3
Deadline of Submission: during the class session on 8 September, 2015
1. Four motors are packaged for sale in a certain warehouse. Each motor sold generates $200
net profit. But a mon
Term 1, 15/16
STAT 201 PROBABILITY THEORY AND APPLICATIONS
Assignment 4
Deadline of Submission: during the class session on 15 September, 2015
1.
Suppose that the cumulative distribution function of the random variable X is given
by
FX (x) =
0
for x < 1
0
Term 1, 15/16
STAT201 PROBABILITY THEORY AND APPLICATIONS
Assignment 11
No need to submit this assignment
1.
You are trying to set up a portfolio that consists of Fund X and Fund Y . Based on the
predicted economic conditions, the joint probability mass f
Term 1, 15/16
STAT201 PROBABILITY THEORY AND APPLICATIONS
Assignment 8
Deadline of Submission: during the class session on 20 October, 2015
1.
Let the joint probability density function of X and Y be
f(x, y) =
1.5y
for 0 < y < 1, y < x < 3y
0
elsewhere
(a
Term 1, 15/16
STAT201 PROBABILITY THEORY AND APPLICATIONS
Assignment 10
Deadline of Submission: during the class session on 3 November, 2015
1.
A manufacturers annual losses follow a distribution with density function:
fX (x) =
2.5(0.6)2.5
x3.5
for x > 0.