Math 2030 A Homework 8
Answers and Solutions
Section 4.1:
9. For i = 1, 2, 3, 4 we have E [Xi ] = 1/2 and V ar[Xi ] = 1/12. Then E [S4 ] = 4 (1/2) = 2 and
V ar[S4 ] = 4 (1/12) = 1/3. Approximate with N (2, 1/3). As S4 has a continuous distribution,
there
We introduce a formal notion of independent events.
Def. 1.46: Two events A and B are said to be independent if
P(A B) = P(A)P(B)
Def. 1.47: The events A1, A2, , An, are said to be pairwise independent
if for any j and k, j k, the events Aj and Ak are ind
Probability
Statistics
What is the difference?
1
Probability is descriptive: formulating (describing) models
Statistics is prescriptive: data collection and analysis of data
Statistics uses the formalism (the models) of Probability
Theoretical Statistics
We introduce a formal notion of independent events.
Def. 1.46: Two events A and B are said to be independent if
P(A B) = P(A)P(B)
Def. 1.47: The events A1, A2, , An, are said to be pairwise
independent if for any j and k, j k, the events Aj and Ak are
ind
Exercise 1.71: Sampling with replacement 25 times from a
population of objects labeled consecutively 1 through N.
Consider the following events:
a. Ai = cfw_ the ith item sampled is 3, i = 1, , 25
b. B = cfw_ the 1st item sampled is either 3 or 5, the 3rd
Example 1.89: Three men and three women are to be seated at
a round table.
(1) What is the number of ways in which the men and woman
can be arranged?
(2) What is the probability that each man is seated between
two women?
(3) Given that each man is seated
F. Distributions and Discrete Density Functions
Def. 1.92: A function f: R R is called a discrete density
function iff
(1) f(x) 0 for all x R
(2) cfw_x R: f(x) 0 is a finite or countably infinite subset of
and is denoted by cfw_x1, x2, x3, xn in the fini
MATH 2030 3.0MW (Winter 2011)
Elementary Probability
Course Outline
Prerequisites:
Single variable integral calculus (MATH 1014 3.0 or MATH 1310 3.0 or equivalent).
Course Webpage
www.math.yorku.ca/~salt/courses/2030w11/2030.html
Lectures:
MWF 8:30-9:20 i
Final exam notes
Thecoursenowswingstowardsstudyingspecificdistributions
andtheirapplications.Alongthewaywelldefineandstudy
meansandvariances.
RecallthatifXhasabinomialdistributionXBin(n,p)
thenP(X=k)=
n
k
_
pk(1p)nk,k=0,1,.,n.
Here0p1andnisapositiveintege
Math 2030 Assignment #2
All page numbers and problems refer to your textbook Pitman. If a problem in the text has more than
one part, hand in only the parts which I ask for. If no parts are stated, then hand in ALL parts. (Example:
3. (a) p. 32 #14 means
YORK UNIVERSITY
Faculty of Liberal Arts and Professional Studies Faculty of Science and Engineering
Midterm
October 14, 2014
Mathematics 2030 A 3.0F
Elementary Probability
NAME: _ _(PLEASE
PRINT)
(Family Name)
(Given Name)
BY SIGNING YOUR NAME, YOU AFFIRM
Math 2030 Solutions to Assignment #1
1.(a) p. 30 # 4a. Yes. If both coins land heads (H) then the number of heads is 2 which is the event
cfw_2. The complement of the event cfw_2 is cfw_0, 1 which corresponds to either the case where not both of
the coins
Math 2030 A Homework 9
Answers and Solutions
Section 3.3:
20.
For (a) observe that the single stock chosen must return a prot of either $ 100 or $ 200 per
$ 1000 invested. The probability this occurs is 1/2.
Let Xi be the prot for the i-th stock. Calculat
York University
Faculty of Science and Engineering
Math 2030 A
Class Test 1
SOLUTIONS
Instructions:
1. Time allowed: 60 minutes.
2. Calculators are permitted.
NO INTERNET CONNECTED DEVICES OR OTHER AIDS PERMITTED
3. Show your work. Your work must justify
York University
Faculty of Science and Engineering
Math 2030 A
Class Test 2
SOLUTIONS
Instructions:
1. Time allowed: 60 minutes.
2. Calculators are permitted.
NO INTERNET CONNECTED DEVICES OR OTHER AIDS PERMITTED
3. A formula list and a table of values fo
Math 2030 A Homework 7
Answers and Solutions
Section 3.2:
5. Let X be
winnings.
XY
0 1
1
1
2
2
3
3
the number of times the gamblers number comes up. Let Y be the corresponding
Probability of occurring
(5/6)3
C3,1 (1/6)(5/6)2
C3,2 (1/6)2 (5/6)
(1/6)3
E [Y
Math 2030 M Homework 1
Answers and Solutions
Section 1.1
2. The sentence has 10 words any one as likely to be chosen as any other. Counting gives
a) 7/10
b) 4/10
c) 4/10
7. Distinguish the dice so that there are 36 equally likely outcomes to a roll of th
Math 2030 M Homework 3
Answers and Solutions
Section 1.5:
3.
P (good | passes) =
P (good)P (passes | good)
.
P (good)P (passes | good) + P (bad)P (passes | bad)
(a) The formula gives
(.8)(1)
40
=
.
(.8)(1) + (.2)(.10)
41
(b) P (bad | passes) = 1 P (good |
Math 2030 A Homework 4
Answers and Solutions
Section 3.1:
9.
P (X = 2) = 1/7 as second must match rst.
P (X = 3) = (6/7) (2/6) as second cant match rst, third must match one of rst two.
P (X = 4) = (6/7) (4/6) (3/5) as second cant match rst, third cant ma
Math 2030 A Homework 5
Answers and Solutions
Section 4.4:
12
1 e 2 z .
2
10.(c) The standard normal variable Z has distribution function (z ) =
This solution purposely avoids using the formula on page 304.
Let Y =
1
(y ) = P (Y y ) = P ( Z y ).
Case y >
Math 2030 M Homework 2
Math 2030 M Homework 2
Answers and Solutions
Answers and Solutions
4. Name the events A and B P (A) = 1 and P B ) = 3. As A and B are independent, so are A and B
4. Name the events A and B . P (A) = .1 and P (B ) = .3. As A and B ar
Math 2030 Assignment #1
All page numbers and problems refer to your textbook Pitman. If a problem in the text has more than
one part, hand in only the parts which I ask for. If no parts are stated, then hand in ALL parts. (Example:
3. (a) p. 32 #14 means