STAT 333 Assignment 1 SOLUTIONS
1. Consider 10 beads on a bracelet having probability p of being blue and probability 1 - p of
being purple. We say a changeover occurs whenever a bead is a different colour from the one
beside it. For example, if the beads
STAT 333 Assignment 1
Due: Friday, Jan 27 at the beginning of class (or up to 5 minutes in)
1. Consider 10 beads on a bracelet having probability p of being blue and probability 1 - p of
being purple. We say a changeover occurs whenever a bead is a differ
STAT 333 Assignment 3
Due: Monday April 2 at the beginning of class
1. At all times, a container holds a mixture of N balls, some white and the rest black. At each
step, a coin having probability p, 0 < p < 1, of landing heads is tossed. If it is heads, a
STAT 333 Quiz .#2 Oct 07, 2013 3:35 4:20 pm
Name: S I.D:
UWUserid: Mark / 20
[5] 1. The joint probability density function of X and Y is given by
-y
f($,y)=67 for 0<x<y and 0<y<oo.
Find E(X2IY = y).
. y i _
Y w For/>0) wax/14m;
#(x ) 7
L902 10(4):) = ji
STAT 333 Quiz #1 Sept 23, 2013 3:35 4:20 pm
Name: 3 LD:
UWUserid: Mark / 20
1. Let X be a discrete random variable having moment generating function (mgf) of the
form ¢X(t) m 1/2+62/3+ 63V 6, t 6 IR. In addition, let Y be another discrete random
variable
_ L0 1,,
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v), F ( Y I 1 j )1 ( l ,
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MW [q :9 Mug? 2' 3/! m 41):} [1 Mg (13,)
STAT 333 Quiz #3 Oct 28, 2013 3:35 - 4:20 pm
Name: S 1.1):
U W Userid: Mark /20
1. Consider a stationary Markov chain {an = 0, 1, 2, . . with transition probability
matrix
0 1 2 3 4
0 0 1/3 2/3 0 0
1 O 0 O 1/4 3/4
P = 2 O 0 0 1/2 1/2
3 0 0 0 0 1
4 1 O 0 O
STAT 333 Quiz #4 Nov 11, 2013 3:35 4:20 p.1n.
Name: S II):
U W Userid: Mark /20
1. Consider the gamblers ruin problem in which Xn represents the gamblers fortune at
time n, n = 0,1,2,., where X E {0, 1,2, . . . , N}. Suppose that X0 = 1 anci N = 4.
Assume
Stat 333: Test 1 - Winter 2012 SOLUTIONS
1
1. Suppose we toss a fair coin (P (H ) = 2 ) until we observe one H , and let Y be the
number of tosses required. Then we toss a second biased coin (with P (H ) = p) until
we observe Y heads, and let X be the num
Stat 333: Test 2 - Winter 2012 SOLUTIONS
1. Suppose an innite sequence of letters is selected randomly from the 26-letter alphabet.
(a) [2] Find the expected # of trials until the rst occurrence of the pattern
EFGEEFGE
E [TEF GEEF GE ] = E [TE ] + E [TEF
STAT 333 Final Exam Review
A
L TEXer:
Properties of binary operations: A(s ) B (s ) =
n
with |s | < min(RA , RB ), A(s )
n=0 (an bn )s
n
n
B (s ) =
with |s | <
n=0
k =0 ak bnk s
min(RA , RB )
W. Kong
Combinatorial identity:
Chapter 1
1
,x A
0
Indicator R
Birth and Death Processes
In many circumstances, it is a dicult task to try and obtain a nice, explicit analytical
expression for Pi,j (t). Mainly for this reason, we often restrict ourselves to certain special
cases of the general continuous-time Markov
Continuous-time Markov Chains
Denition: A stochastic process cfw_X (t), t 0 is said to be a continuous-time MC if the
following conditions hold true:
(i) For t 0, X (t) takes on a nite (countable) set of possible values (we assume that
the general state s
Let Y1 , Y2 , . . . , Yn be an i.i.d. sequence of random variables having a common continuous
distribution on (0, ) with c.d.f. F (y ) = P (Yi y ) and p.d.f. f (y ) = F (y ) for any
i = 1, 2, . . . , n.
Dene:
Y(1) 1st smallest among cfw_Y1 , Y2 , . . . ,
STAT 333 Quiz #5 Nov 25, 2013 3:35 4:20 pm
Name: S I.D:
UWUserid: Mark / 20
In this quiz, you may nd the following facts useful:
0 If X N Exponential(/\), then P(X > :17) = _)I for x Z 0.
o If X ~ Gamma(n, A), then P(X > w) = 6" ZZOIQCEY/il for x 2 0.
o
STAT 333 notes: Applied Probability
Johnew Zhang
November 29, 2012
Contents
1 Introduction
1.1 Basic Concepts of Probability . . . . . . . .
1.1.1 What is a probability model? . . . .
1.2 Review of Random Variable . . . . . . . . .
1.3 Some important dist