Independence of two random variables
For two events (called A and B), we defined
independence (in Lecture 6) as follows:
P (A B) = P (A)P (B)
so that
P (A B)
= P (A)
P (A|B) =
P (B)
i.e. being given that B has happened does not
change the probability one
Multi-variate Continuous Distributions
Two (or more) random variables are jointly
continuous when they have a joint
cumulative distribution function which is
continuous in both variables:
x y
F (x, y) =
f (s, t)dt ds
The function f (x, y) is the joint pr
STM2PM and STM4PM
ASSIGNMENT 5, 2017
Place your assignment solutions in the appropriate box on the third level of the Mathematics
and Statistics Building (Physical Sciences 2) before 1 pm on Thursday 12 th October.
In submitting your work, you are consent
Question (8): Assume that the probability to pass in a course in Arabic is 60% and the
probability to pass in a course in English is 45%, also the probability to pass in both courses is
20%. If a student takes both courses, then determine the sample space
P
STM2PM
PRACTICE CLASS 10
Joint Distributions
When two random variables are defined on the same sample space, their probability
properties can be described by
a joint probability mass function pXY (x, y), if they are both discrete
a joint probability d
Question 1 :
Its well known that I like red jelly beans. Less well known is the fact that I
dont like purple ones. In a packet of 20 jelly beans, 3 are red and 4 are
purple. Use the multivariate hypergeometric distribution to calculate the
probability tha
Times.
Lectures: MW 11:30 AM - 12:45 PM, Math & Science Center, Room
W201
Lab: F 11:30 AM - 12:20 PM, Chemistry, Chem 260
First day of classes: August 25, 2016
Last day of classes: December 6, 2016
Recess: Sept 04 (Labor Day), Nov 22+24 (Thanksgiving)
Off
THE CALCULUS EXAM 2 STUDY GUIDE
Ideas involved, skills needed, types of problems you will see (all of the following will appear
on the exam):
Equivalent notations for the derivative of = ():
' =
=
= ' .
THE CALCULUS EXAM 1 STUDY GUIDE
Ideas involved, algebra and arithmetic skills needed, and types of problems that will appear:
!"#$
Find the slope
Know how to use F.O.I.L. Be familiar with polynomials and related
4 INDEPENDENCE (SECTION 2.5)
Name(s):
For this worksheet, it may be helpful to recall, Geometric series: for 0 < a" < 1 we have
1. For the following blood type percentages, what is the probability that a couple has dili'erent
blood types?
Blood type
Math 361: Section 2.5
2.5.14 In a roll of a pair of fair dice (one red and one green), let A be the event the red die
shows a 3, 4, or 5; let B be the event the green die shows a 1 or a 2; and let C be the
event the dice total is 7. Show that A, B, and C
Math 361: Homework Section 2.4
2.4.6 Two events, A and B, are defined on a sample space S such that P(A|B) = 0.6, P(At least
one of the events occurs) = 0.8, and P(Exactly one of the events occurs) =0.6. Find P(A)
and P(B).
We are given:
P (A|B) =
P (A B)
2 SAMPLE SPACES 85 PROBABILITY (SECTlONS 2.22.3)
Name(s):
1. For each of the following experiments below give the set that describes the event given.
(a) Experiment: deal 5 cards from a. 52 card deck
Sample Space:
S = cfw_the 2, 598, 960 possible 5 card
3 CONDITIONAL PROBABILITY (SECTION 2.4)
Name(s):
1. Consider families with 2 children and assume that having a B or G is equally likely.
Q0: What is the sample space?
Q1: What is the probability of two girls given that at least one child is a girl? Find
2016 Global Industry 4.0 Survey
What we mean by Industry 4.0 / Survey key findings / Blueprint for digital success
Industry 4.0: Building
the digital enterprise
2000+
respondents in 26 countries
US$493 bn
in digital revenue gains p.a.
US$421 bn
p.a. in co
2
2. August 30, sets, functions and graphs
2.1. Basics. A set is a collection of elements of interest without self-contradiction.
Let be the empty set. Let R be the set of real numbers.
We say an element a is in a set A and write a A, if a is an element o
R E F E R E N C E PA G E 1
Cut here and keep for reference
ALGEBRA
GEOMETRY
Arithmetic Operations
Geometric Formulas
a
c
ad bc
b
d
bd
a
d
ad
b
a
c
b
c
bc
d
ab c ab ac
a
c
ac
b
b
b
Formulas for area A, circumference C, and volume V:
Triangle
Circle
S
Math 1114951; Calculus I
Sample Midterm Exam I
W
Name: ii: 2;
9 problems, 6 pages, 37 total points. Read the instructions to each question carefully. Calculator
is not allowed. The Honor Code is in effect.
Show your work. Illegible handwriting or irrele
Math 111-011 Calculus I
Sample Midterm Exam I
Name:
9 problems, 6 pages, 37 total points. Read the instructions to each question carefully. Calculator
is not allowed. The Honor Code is in effect.
Show your work. Illegible handwriting or irrelevant works r
Math 111-011 Calculus I
Sample Midterm Exam II
Name:
8 problems, 4 pages, 40 total points. Midterm II consists of 24% of your weighted total grade.
Read the instructions to each question carefully. Calculator is not allowed. The Honour Code is in
effect.
1. August 28
In this lecture, we read the syllables and did a high school algebra problem
9(8 x) 8 x 11 x
+
+
=x
(9 8)x 9 8
x1
First, simplify get x3 42x + 36 = 0, observe that one root is 6, then by division
3
(x 42x + 36)/(x 6) = x2 + 6x 6 = 0, by quadr
4
Test 8 (The vertical line test). A subset of the xy-plane is the graph of a function
if no vertical line intersects the curve more than once.
Proof. Denote the subset C R2 , if x = a is a vertical line such that it intersects C
more than once, i.e. at l
18
10. September 20, Continuity
Theorem 96. If f (x) = g(x) when x 6= a, then lim f (x) = lim g(x).
xa
xa
Example 97. f (x) = x if x 6= 0 and f (x) = 1 when lim f (x) = 0 not 1!
x0
Example 98. If f (a) = g(a), it is not necessarily true that lim f (x) = l
12
7. September 13, examples of inverse functions
Proposition 62. The graph of f 1 is cfw_(x, y) | y = f 1 (x) = cfw_(x, y) | x = f (y).
a+b
The middle point of the line interval defined by (a, b) and (b, a) is ( a+b
2 , 2 ), which
is on the line y = x. A
16
9. September 18, Infinity
9.1. One-sided limits.
Definition 78. We say lim f (x) = L if f (x) approaches L as x approaches a
xa
from the left. We say lim+ f (x) = L if f (x) approaches L as x approaches a from
xa
the right.
Proposition 79. lim f (x) ex
7
4. September 6, Dynamic properties of a function, trigonometric
functions
New functions from old functions
4.1. Transformation of functions. Let Y = f (X) be a function. We transform X to x by x = g(X) and we transform Y to y by y = h(Y ), then
y = h(Y
11
6. September 11, 2013, Inverse functions
Definition 53. A function f is a one-to-one function if x1 6= x2 imlies that
f (x1 ) 6= f (x2 ). Or equivalently, f (x1 ) = f (x2 ) implies that x1 = x2 .
f is not 1-1 if there are x1 6= x2 such that f (x1 ) = f