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 wr
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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
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 =
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
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 p
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 seate
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.
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
independen
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 formal
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 subs
MATH 2030 3.00 A S1 2013
Solutions to HW Problems for Sections 1.4, 1.5 and 1.6.
May 19, 2013
Section 1.4
Exercise 4.
Solutions:
We are given P (A) = 0.1, P (B) = 0.3, P (AB) = P (A)P (B).
(a) P [(A B
MATH 2030 3.00 A S1 2013
Solutions to HW Problems for Appendix 1 and Sections 1.1, 1.3.
May 16, 2013
Appendix 1
Solutions:
Exercise (vii)
Sum of 0s and 1s equals k if there are exactly k 1s and n k 0s
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 MATHEMATICA are arranged in a random
Formulae
P (AB)
.
P (B)
P (A) = n P (A | Bi )P (Bi ).
i=1
P (A | Bi )P (Bi )
.
P (Bi | A) = n
i=1 P (A | Bi )P (Bi )
P (A1 A2 An ) = P (A1 )P (A2 | A1 )P (A3 | A1 A2 ) P (An | A1 A2 An1 ).
For X Bin(n
Probability
Assignment 1
Rahul Maini
210205623
Professor Ray N.-R. Shieh
MATH 2030 Section A
Wednesday, September 21, 2011
Section 1.1 #2 page 9:
(a) 7 out of 10 words have at least 4 letters, thus th
YORK UNIVERSITY
Faculty of Liberal Arts and Professional Studies Faculty of Science and Engineering
Class Test I
October 5, 2010
Mathematics 2030 A 3.0F
Elementary Probability
NAME: _ _(PLEASE PRINT)
York University
MATH 2030 3.0 (Elementary Probability) Fall 2011
Assignment 1 Solutions, Sept 2011
1.1 No. 2
(a) We use an equally likely outcomes model with
= cfw_suppose, a, word, is, picked, at, r
York University
Faculty of Science and Engineering
MATH 2030 3.0AF 2011F Elementary Probability
Instructor: Ray N.-R. Shieh
Second Midterm Examination
Nov. 22, 2011
NAME(in print):
SIGNATURE:
STUDENT
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)
(Famil
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 paramet
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.
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 prod
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 /
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
Solutions to Assignment #5
1 (a) Pitman (a)p. 91 11a) [You need to know that for B(n,p) the mode m =int(np+1) where int denotes
the integer portion of the number. See p.86 Pitman. Here is how the resu
Theorem 2.10:
(1) If X is a discrete real valued random variable, then
f(x) P(X = x) is a discrete density function
(2) If f is a discrete density function, then there exists a r.v. X whose
density is
An important identity:
(z) = 1 (z)
Picture:
Consider
P(a < X b) = P(X b) P(X a)
= (b) (a)
Notation:
We define
(a, b) (b) (a)
The symmetry of the density implies that
( z) = 1 (z)
(1)
which means that
Example 2.50: (Multinomial distribution) Consider
repeatedly and independently rolling a biased die 20
times. On each roll, the possible outcomes are 1, 2, 3, 4,
5, 6 with probabilities p1, p2, , p6.
MATH 2030 (A) TEST#1 FALL 2017
YORK UNIVERSITY
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS AND STATISTICS
ELEMENTARY PROBABILITY (MATH 2030)
TEST #1
OCTOBER 10, 2017
Instructor: Dr. Igor Poliakov -
DEPARTMENT OF MATHEMATICS AND STATISTICS
YORK UNIVERSITY
COURSE OUTLINE
MATH 2030 3.0 A, FALL 2017
ELEMENTARY PROBABILITY
Section A Wednesday, 7-10 pm, CLH D
Text book: Probability by Pitman, Springer
York University
Department of Mathematics and Statistics
2016 - Fall Term
Math 2030: Elementary Probability
Test-1
Exam Date: October, 4, 2016
Instructor: Gkhan Yildirim
Last name:
First name:
York ID
York University
Department of Mathematics and Statistics
2016 - Fall Term
Math 2030: Elementary Probability
Test-3
Exam Date: November, 29, 2016
Instructor: Gkhan Yildirun
Last name: 0
First name: _*_