MATH 3D03
Short Answers to Test # 2
# 1 (6 marks) State whether the following statements are true or false.
For any two events E and F in a sample space:
(i) P (E) + P (F ) = P (E) + P (F ) P (E F )
(ii) P (E|F ) = P (E F |F )
typo, so everyone gets full
Math 3D03
M. Min-Oo
Assignment #2
Due: Tuesday, February 3rd, 2015 in class (at the beginning of the lecture period)
Note: You can use symbolic software only to check your answers (for the integrals for example)
but you are required to show your calculati
Math 3D03
Assignment #3
Due: Thursday, February 26th, 2015 in class
Note: You can use symbolic software to check your answers (for the integrals for example) but you
are required to show your calculations
1.
Show that
w = tan(z)
maps the vertical strip |x
Math 3D03
Assignment #1
Due: Tuesday, January 20th, 2015 in class (please hand it to me at the beginning
of the lecture period)
Note: You are required to show your calculations. You can use symbolic software only to check
your answers.
1.
Compute all valu
Math 3D03
Short solutions to assignment #5
1. If X1 , X2 , X3 are independent and identically distributed exponential random variables with the
same parameter > 0, compute the probability
Pcfw_max(X1 + X2 , X3 ) 2
X1 + X2 is a Gamma distribution with = 2,
Asymptotics of the Airy function
1. Denition:
Ai(z) =
+
1
2
exp(i(
+
k3
1
+ z k) dk =
3
cos(
0
k3
+ z k) dk
3
Changing variables: s = ik and choosing the right contour C to integrate, we can also write this
as:
s3
1
exp( + z s) ds
Ai(z) =
2i C
3
The conto
Math 3D03: Term Test # 1
February 10, 2005
FAMILY NAME:
GIVEN NAME(S):
STUDENT NUMBER:
SIGNATURE:
Instruction: No aids allowed, except for the McMaster standard calculator Casio
fx-991. The duration of this test is 50 minutes.
This test has 5 questions, e
Math 3D03: Term Test # 2
March 10, 2005
FAMILY NAME:
GIVEN NAME(S):
STUDENT NUMBER:
SIGNATURE:
Instruction: No aids allowed, except for the McMaster standard calculator Casio
fx-991. The duration of this test is 50 minutes.
This test has 5 questions, each
Math 3D03
Assignment #5
Due: Tuesday, March 31st, 2015 in class
Note: You can use symbolic software to check your answers but you are required to show your
calculations
1. If X1 , X2 , X3 are independent and identically distributed exponential random vari
Math 3D03
Assignment #3
Due: Thursday, February 26th, 2015 in class
Note: You can use symbolic software to check your answers (for the integrals for example) but you
are required to show your calculations
1.
Show that
w = tan(z)
maps the vertical strip |x
Math 3D03
Assignment #4
Due: Tuesday, March 10th, 2015 in class
Note: You can use symbolic software to check your answers (for the integrals for example) but you
are required to show your calculations
1. If 5 indistinguishable marbles are placed at random
Math 3D03
Short solutions to assignment #2
1.
Evaluate the following denite (real-valued) integrals:
2
(sin )n d
(i)
for n N.
What happens when n ?
0
(ii)
(iii)
0
eax
dx
1 + ex
dx
1 + xn
for 0 < a < 1
where n 2 is an integer
(i) The integral is obviously
Math 3D03
Assignment #5
Due: Tuesday, March 31st, 2015 in class
Note: You can use symbolic software to check your answers but you are required to show your
calculations
1. If X1 , X2 , X3 are independent and identically distributed exponential random vari
Math 3D03
Short solutions to assignment #3
1.
Show that
w = tan(z)
maps the vertical strip |x| <
4
in the z-plane onto the unit disk |w| < 1 in the w-plane.
Write w = tan(z) as a composition of two maps W = i e2iz and w = W i . The rst transformation
W +i
MATH 3D03
TEST # 1
# 1.
Evaluate the following complex contour integrals:
C
z dz
(z 3)2 (z + 1)
where C is the circle dened by: |z| = 2 (taken counterclockwise)
1
One simple pole at z = 1 inside the contour with residue = 16 . Therefore the answer is i .
Math 3D03
Short answers to assignment #5
1. Do problem 31.4 on page 1298 in the textbook.
Use a linear regression (least squares t) of the form Y = + X, where we can choose:
x
1
either X = x, Y = y , = b, = a
or
X = x , Y = x , = a, = b
y
The best estimat
Math 3D03
Assignment #5
Due: Monday, April 3rd, 2013 in class (at the beginning of the lecture period)
You can use statistical software to check your answers but you are required to show your calculations
1. Do problem 31.4 on page 1298 in the textbook.
2
Math 3D03
M. Min-Oo
Assignment #4
Due: Wednesday, March 20th, 2013 in class (at the beginning of the lecture period)
1.
If you throw a cubical and fair die (singular of dice!) 6n times, what is the probability of
getting each face exactly n times. Use Sti
Math 3D03
M. Min-Oo
Short Answers to Assignment #4
1.
If you throw a cubical and fair die (singular of dice!) 6n times, what is the probability of
getting each face exactly n times. Use Stirlings formula to nd out what happens when n .
The number of ways
Math 3D03
M. Min-Oo
Assignment #3
Due: Wednesday, February 27th, 2013 in class (at the beginning of the lecture
period)
1. (10 marks) The complex potential
(z) = z +
1
i log(z)
z
where is a positive real number, describes a uid ow around a cylinder with
Math 3D03
Short Answers to Assignment #3
1. The complex potential
1
i log(z)
z
where is a positive real number, describes a uid ow around a cylinder with circulation. Locate
the stagnation points (as a function of ) and sketch the streamlines of the ow,
Math 3D03
Short solutions to assignment #1
i
1. Compute all values of i(i ) and (ii )i
All the values of ii are given by: exp(i log(i) = exp(i(i( + 2k) = exp( 2k)
2
2
kZ
so
i
(i) i(i ) = exp(ii (log(i) = exp ie 2 2k ( + 2l)
2
ii
(ii)
i
k, l Z
= exp(i(log(
Math 3D03
Assignment #1
Due: Wednesday, January 23rd, 2013 in class (at the beginning of the lecture
period)
Note: You are required to show your calculations. You can use symbolic software only to check
your answers.
1.
Compute all values of
i
i(i )
2.
an
Math 3D03
Short solutions to assignment #2
1. Evaluate the following denite (real-valued) integrals:
(i)
2
n
0 (sin ) d
for n N.
What happens when n ?
The integral is obviously zero for odd n, since sin() = sin(2 ). For even n, we have using
the binomial