MATH 2030 Elementary Probability! Winter 2014
Test 1! Feb. 3, 2014
Student Name: ID-No.:
You have 50 minutes to solve the following problems: Show your complete work.
Permitted aids: calculator and writing utensils; 1 written help-sheet of standard
letter
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YORK UNIVERSITY
Faculty of Science
Department of Mathematics and Statistics
MATH 2030 3.00 A
Test #1
May 23, 2013
Solutions
1. (4 + 4 pts) The letters of the word STATISTICS are arranged in a random order.
Find the pro
York University MATH 2030 3.0AF (Elementary Probability) Assignment 4 Solutions October 2008 Salisbury 2.1 No. 6 (a) Let X be the number of shots that hit the bullseye. Then X Bin(8, 0.7) so P (X = 4) = 8 (0.7)4 (0.3)4 = 70 0.2401 0.0081 = 0.136
York University MATH 2030 3.0AF (Elementary Probability) Assignment 5 Solutions November 2008 (postponed to February 2009) Salisbury 3.2 No. 8 E[(X +Y )2 ] = E[X 2 +2XY +Y 2 ] = E[X 2 ]+2E[XY ]+E[Y 2 ] = 3+22+4 = 11 3.2 No. 14 Let X be the number o
Final Exam of MATH 2030
The weight is 45%. The Final Exam is at Dec 11 Sunday, 7:00-9:30 pm, SLH F.
Content: the material covered in class and on the assignments .
This is roughly Chapters 1-4 of the text, plus Appendix 1 and a little bit from Section 6.4
Math 2030 2014 Solutions to Assignment #3
.
Note: What Pitman refers to as "joint distribution tables" are really "joint density tables". Just as we
called fX(x) = P(X = x) the density of a discrete r.v. X, fX,Y(x,y) = P(X = x, Y= y) ("X= x and Y = y") is
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
Solutions to Assignment # 4
1.(a)
FX,Y(x, y) = P(X x, Y y) To find the marginal cumulative distribution function (cdf)for X, we
let y increase to + ; i.e. lim FX,Y ( x, y ) FX ( x ) . This means that the last column of the joint
y
cumulative distribution
Math 2030 Solutions to Assignment #2
1 (a) A bookstore has a draw for a $25 book certificate. Each customer draws a ticket from a bowl of
100 tickets. Only one of the tickets is a winning ticket and once the certificate is won, the draw is over. If
the ti
Solutions to Assignment #6
1 (a) Pitman p.121 #2
Let N = # of successes in 500 independent trials with probability of success on each trial p =.02
Then N is B(500, .02), that is, binomial with parameters n= 500, p = .02
We note that np2 = 500 (.02)2 = .2
YORK UNIVERSITY
Faculty of Liberal Arts and Professional Studies Faculty of Science and Engineering
Midterm
June 11, 2015
Mathematics 2030 A 3.0S1
Elementary Probability
NAME: _ _(PLEASE
PRINT)
(Family Name)
(Given Name)
BY SIGNING YOUR NAME, YOU AFFIRM T
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
Math 2030 Material Required for Final Exam 2015
Lecture Notes: all posted handouts
From your text (Pitman):
All sections listed in Course Outline
Make sure that you understand:
(1) Outcome space and probability space
(2) partition and disjoint events
(3)
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 finit
Math 2030-2015 Assignment #5
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 m
Math 2030 Material Required for Midterm Exam 2015S1
Lecture Notes: all posted handouts for Section 1
From your text (Pitman):
Chapter 1: All except for odds.
Appendix 1: All
Appendix 2
Chapter 2: Section 1: up to page 85 inclusive; Section 2.1; 2.5 p.123-
Def. 1.41: If an outcome space O has finitely many elements a1, a2,
, an, then the outcomes are said to be equally likely or
equiprobable iff P(cfw_a1) = P(cfw_a2) = =P(cfw_an).
For finite outcome spaces, we will use the term at random or
randomly to mean
Math 2030 Solutions to Assignment #2
1 (a) A bookstore has a draw for a $25 book certificate. Each customer draws a ticket from a bowl of
100 tickets. Only one of the tickets is a winning ticket and once the certificate is won, the draw is over. If
the ti
Math 2030 2015Solutions to Assignment #3
.
Note: What Pitman refers to as "joint distribution tables" are really "joint density tables". Just as we
called fX(x) = P(X = x) the density of a discrete r.v. X, fX,Y(x,y) = P(X = x, Y= y) ("X= x and Y = y") is
E. Two- Stage Process and Bayes' Rule
Tree Diagram for Two Stage Process:
1st stage: possible outcomes
A1, A2, , A10
2nd stage: possible outcomes
B1, B2, , B5
So what is the tree diagram?
88
(Pitman p.48) Consider a two stage process where B1, B2, Bn is a
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 finit
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 in
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
I. Geometric Distribution
1. In a success failure experiment, we want the waiting
time until the first success
Example 2.43: An auto parts manufacturer has found that
its defect rate is 3% of its production. If the parts are
coming off an assembly belt, w
F. Expectation of a Continuous Random Variable
For X ~ U(a, b), what is E(X) ?
One approach:
Partition (a,b) starting at a and ending at b into n intervals
each of size (b a) / n.
Put probability 1 / n to the set of mid-points of of the n
intervals
cfw_a+
C. (Cumulative) Distribution Functions (cdf)
Def. 2.18: A function F: R R is called a (cumulative) distribution
function (c.d.f) if
(i)
F(x) 0 for all x R;
(ii)
F is increasing, that is, if a < b, F(a) F(b)
lim
(iii) x F ( x ) = 1 and x lim F ( x ) = 0
(i
Commonly Tested Discrete Distributions
Commonly Tested Discrete Distributions
Standard Discrete Distributions We will use the shortcut notation
f X ( x ) = Pr( X = x ) = p x .
1. X ~ U ( N ) (Uniform Distribution on the set cfw_1,2,N.)
X = the outcome of
Expectation and Other Parameters
Page 1 of 3
Expectation and Other Parameters
Expectation (denoted E[ X ], X , or ) For a random variable X, the
expectation of X (aka expected value of X, or mean of X) is the weighted
average of the values of supp(X). The
Review of Algebra, Calculus, and Counting Techniques
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Review of Algebra, Calculus, and Counting Techniques
Set Theory The typical question can be solved using Venn Diagrams,
which well illustrate by example.
Inverse Functions The inverse of the
Additional Important Facts Involving Random Variables
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Additional Important Facts Involving Random Variables
Transformations
(Discrete Case) For a transformation of a discrete random variable, write
out the probability distribution table.
(Conti
Random Variables
Page 1 of 2
Random Variables
Random Variables A random variable is a process, which when followed,
will result in a numeric output. The set of possible outputs is called the
support, or sample space, of the random variable. Associated wit