MATH 2131 TEST TWO
Friday, March 2; 50 minutes
NAME:
STUDENT NUMBER:
There are 4 questions worth a total of 45 points. Note that a more difcult question is not always
worth more points. Use your time wisely.
You may use the back of the pages if you run ou
MATH 2131 QUIZ 1
Monday, January 17; 15 minutes
NAME:
STUDENT NUMBER:
MAKE SURE TO EXPLAIN YOUR WORK FULLY!
(1) (4 marks) Prove Bonferronis inequality: P (A B ) P (A) + P (B ) 1.
(2) Suppose that X has density function f (x) = cx3 for 0 x 1 and f (x) = 0
MATH 2131 3.00 MS2
Assignment 1
Total marks = 40
Question 1: Let (X, Y ) have the joint probability density function
fX,Y (x, y ) =
cx/y 2 , y 1 < x < y,
0,
otherwise.
1 < y < 2,
(a) (4 marks). Find c such that fX,Y (x, y ) is a joint probability density
MATH 2131 FINAL EXAM
Thursday, April 21st; 180 minutes
NAME:
STUDENT NUMBER:
READ THE FOLLOWING CAREFULLY AND SIGN BELOW:
This exam has a total of 12 questions worth 150 points and is written on 13 pages (including this
page). Use your time wisely and do
MATH 2131 MIDTERM TEST
Friday, February 3; 50 minutes
NAME:
STUDENT NUMBER:
There are 5 questions worth a total of 45 points. The rst three questions are worth 32 points.
Use your time wisely.
The last page contains some densities which you may nd useful.
MATH 2131 3.00 MS2
Assignment 3
Total marks = 50
Question 1: Let (X, Y ) have the joint pdf
fX,Y (x, y ) =
ey , 0 < x < y < ,
0,
otherwise.
(a) (4 marks). Find E (XY ).
(b) (6 marks). Find Cov(X, Y ).
(c) (4 marks). Find X,Y .
Question 2: Let X1 , . . . ,
MATH 2131 TEST THREE
Friday, March 30th; 50 minutes
NAME:
STUDENT NUMBER:
There are 5 questions worth a total of 40 points.
You may use the back of the pages if you run out of space.
A separate page with formulas and the necessary tables is provided.
If a
MATH 2131 3.0 - Winter 2016
Assignment 3
(Due Date: March 02, 2016)
Only hand in the questions with
Question 1 :
Let X Geometric(p). Find the mean, variance, moment generating
function, and probability generating function of X.
Question 2:
Let X Exponent
MATH 2131 MIDTERM TEST
Friday, February 18; 50 minutes
NAME:
STUDENT NUMBER:
(1) Let X Exponential() and recall that the cdf of X is
F (x) =
0
x < 0,
1 ex x 0.
(a) (5 marks) Find the cdf of Y = min(1, X ).
(b) (3 marks) Is Y discrete or continuous?
(2) Le
MATH 2131 QUIZ 4
Monday, March 21st; 20 minutes
NAME:
STUDENT NUMBER:
MAKE SURE TO EXPLAIN YOUR WORK FULLY!
(1) (5 marks) A fair coin is tossed n times, and the number of heads, N, is counted. The
coin is then tossed N more times. Find the expected number
MATH 2131 3.0 - Winter 2017
Assignment 1
(Due Date: Feb 10, 2017)
Question 1: Let X Geometric(p). Show that P (X > n +k 1|X > n 1) = P (X > k).
Question 2: The gamma function is defined by
(p) =
Z
et tp1 dt.
0
Show that
a. (1) = 1.
b. (1/2) =
.
c. Let n
MATH 2131 3.0 - Winter 2016
Solution for Assignment 3
Question 1: Given that X Geometric(p), we have p(x) = p(1 p)x1
x = 1, 2, 3, .
E(X) = p + 2p(1 p) + 3p(1 p)2 + 4p(1 p)3 +
(1 p)E(X) = p(1 p) + 2p(1 p)2 + 3p(1 p)3 +
By taking the difference of the abo
MATH 2131 3.0 - Winter 2016
Solution for Assignment 2
Question 1:
1
xm/21 ex/2
m/2
2 (m/2)
1
fY (y) = n/2
y n/21 ey/2
2 (n/2)
1
xm/21 y n/21 e(x+y)/2
f (x, y) = (m+n)/2
2
(m/2)(n/2)
fX (x) =
Let W =
Therefore
X/m
Y /n
=
nX
mY
, and U = X, we have X = U ,
MATH 2131 3.0 - Winter 2016
Solution for Assignment 1
Question 1:
a. F () = 0, F (+) = 1 and as x increases, expcfw_x decreases and hence F (x)
increases. Thus F (x) is an increasing function of x. So, F (x) is a cdf.
dF (x)
dx
b. f (x) =
= x1 expcfw_x .
MATH 2131 3.0 Section M
Test 2 Solution
Question 1:
a. X1 , , Xn iid N (, 2 )
Hence
Thus
X1
2
,
Xn
2
n
X
Xi
i=1
X1
, , Xn
iid 21 .
2
iid N (0, 1).
n
1 X
= 2
(Xi )2 = Y 2n .
i=1
b. We have var(Y ) = var[E(Y |X)] + E[var(Y |X)].
Therefore E[var(Y |X)]
MATH 2131 3.0 - Winter 2016
Solution for Assignment 4
Question 1: Given that X Geometric(p), we have E(X) = p1 and var(X) = 1p
.
p2
1p
1
Let X1 , . . . , Xn iid Geometric(p), then E(X) = p and var(X) = np2 . Furthermore, let
Xi 1/p
Zi = q
(1 p)/p2
we have
MATH 2131 3.0 - Winter 2016
Assignment 4
(Due Date: March 23, 2016)
Only hand in the questions with
Question 1 : Assume X1 , . . . , Xn be iid Geometric(p). Without using the Central Limit
XE(
X)
Theorem, show that the asymptotic distribution of
is the
MATH 2131 3.0 Section M
Test Solution
Question 1:
a. We have
And let W = U 2
fU (u) = 1
dU
=
U = W dW
(
fW (w) =
0 u 1.
1 .
2 W
1
2 w
Therefore,
0w1
else
0
b.
F (x) =
Z x
h
t1 dt = t
1
Therefore
(
F (x) =
ix
1
= 1 x
x > 1.
x1
x>1
0
1 x
Question 2:
a. The
Example:
Theres only two possible outcomes of any military operation - Success or Failure. For
each military operation, the US government assumed that the probability of a success is
p. US carried out one military operation against ISIS yesterday. Let X b
MATH 2131 FINAL EXAM
Thursday, April 12th; 180 minutes
NAME:
STUDENT NUMBER:
READ THE FOLLOWING CAREFULLY AND SIGN BELOW:
This exam has a total of 11 questions worth 140 points and is written on 11 pages (including this
page). Use your time wisely and do
MATH 2131 QUIZ 2
Monday, January 31; 15 minutes
NAME:
STUDENT NUMBER:
MAKE SURE TO EXPLAIN YOUR WORK FULLY!
(1) (5 marks) Suppose that X exp() and let Y = max(2, X ). Find the cdf of Y .
(2) (5 marks) Let (X, Y ) have joint density
f (x, y ) =
1
(x2
8
y