Surname
Given Name
Student Number
Queens University
Department of Mathematics and Statistics
STAT/MTHE 351
Midterm Examination October 23, 2013
Total points = 36. Duration = 2 hours.
Closed book, closed notes.
Simple calculators are permitted.
Write t
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 4
1. [Suppose events A, B, and C are. . . ]
We have to show that the independence of A, B, and C implies P (A(B C) = P (A)P (B C).
We can write
P (A(B C) = P (AB AC) = P (AB) + P (AC) P (ABAC)
= P
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 3
1. [Section 3.1 # 4.]
The sum of the two dice is divisible by 5 if and only if it is either 5 or 10. Let A be the event that
the sum is divisible by 5 and B that both dice show 5. Taking the samp
STAT/MTHE 351 Fall 2013:
Solutions to Assignment 5
1. (a) [Ghahramani, 4.2, # 1] The set of values that X can take is given by
cfw_|i j| : i, j = 1, 2, 3, 4, 5, 6 = cfw_0, 1, 2, 3, 4, 5.
We now determine the probabilities associated with these values. Not
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 2
1. [A real number x is selected. . . ]
(a) The sample space S is the interval [2, 3]. Since x is always in this interval we can express the
events A, B, and C as subsets of [2, 3]:
A = cfw_x [2,
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 10
1. [Ghahramani, 8.2, # 16] Let Z = max(X, Y ). For any t we have that max(X, Y ) t if and
only if X t and Y t. Thus by independence of X and Y ,
1
FZ (t) = P (max(X, Y ) t) = P (X t, Y t) = P (X
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 8
1. [Ghahramani, 7.1, # 7]
The problem can be rephrased as follows: Let X be a uniform random variable on the interval [0, ].
Find the probability
P X 3, X 3 .
We have
P X 3, X
3
= P X 3, X
2
3
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 9
1. [Ghahramani, 8.1, # 2] We have the joint pmf of X and Y given to be
c(x + y) if x = 1, 2, 3, y = 1, 2
0
otherwise
p(x, y) =
where c is a constant.
(a) To nd the correct value of c, we use the
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 6
1. [Let X be a random variable . . . ]
Recall that a random variable is a constant with probability 1 if and only if its variance is zero. We
will show that since the random variable X given to u
STAT/MTHE 351 Fall 2013:
Solutions to Homework Assignment 7
1. [Consider the function. . . ]
(a)
f (x) =
k (2x x3 )
0
if 0 < x < 3/2
otherwise.
Now, f (x) can be a probability density function (pdf) only if it is non-negative for all values
of x. Now, obs