What is Probability?
Quantification of uncertainty. Mathematical model for things that occur randomly. Random not haphazard, don't know what will happen on any one experiment, but has a long run order. The concept of probability is necessary in
STATISTICAL LABORATORY, March 22nd, 2011
CONDITIONAL PROBABILITY AND
INDEPENDENCE
Manuela Cattelan
1
SIMPLE PROBLEMS
Ex1 A couple has two children. 1) What is the probability that both are girls, given that
the oldest is a girl? 2) What is the probability
STA257
PROBABILITY
Ch 3, Joint Distributions, Part 2
Instructor
Dragan Banjevic
Fall 2016
1
Independent Random Variables (I)
(Why in red? Because the independence is very important, it is
one of the key notions in Probability. Then, pay attention!)
Whe kn
STA257
PROBABILITY
Ch 5, 6 Limit Theorems
Instructor
Dragan Banjevic
Fall 2016
1
The Law of Large Numbers (I)
We built our theory hoping to get a good model for real
phenomena of random experiment, random events, and their
probabilities. Our main interpre
STA256H5F
Final Exam Review
(Part III Continuous Random Variables)
Part I Continuous Random Variables
Continuous Random Variables
-
A random variable
Y
is called continuous if it can take any value within a finite or infinite
interval of the real line.
-
STA256H5F Final Review Practice Problems Part IV
1. Let
Y 1 and Y 2 have the joint probability density function given by
cfw_
f ( y 1 , y 2) = k ( 1 y 2 ) , 0 y 1 y 2 1,
0,elsewhere .
k that makes this a probability density function.
b) Find P(Y 1 3/ 4,Y
STA256H5F Final Review Practice Problems Part III
Y
1. A random variable
has the following distribution function:
cfw_
0, for y <2,
1/8, for 2 y <2.5,
3 /16, for 2.5 y <5,
F ( y )=P ( Y y ) = 1/2, for 4 y <5.5,
5 /8, for 5.5 y <6,
11/16, for 6 y <7,
1, fo
STA256H5F
Final Exam Review
(Part II Discrete Random Variables)
Part I Discrete Random Variables ()
Definition
A random variable Y is said to be discrete if it assumes only a finite or countably infinite number
of distinct values.
1
Part II Discrete Pr
STA256H5F
Final Exam Review
(Part I Probability)
Part I Set Operations
The union of A and B (A B) is the set of all points in A or B or both, denoted by
A B
A B=cfw_ x
S: x
A or x
B
Venn Diagram
The intersection of A and B (A B) is the set of all point
STA256H5F Final Review Practice Problems Part II
1. A group of four components is known to contain two defectives. An inspector tests the
components one at a time until the two defectives are located. Once she locates the two
defectives, she stops testing
STA257: Probability and
Statistics I
Section L0101 Fall 2015I
Week 2, Lecture 4
1
Partitions
Definition: A partition of is a finite collection of
events cfw_B1,B2,.,Bn, such that
Bi
Bj
n
, i
B2
j
A
B1
UB
i
B3
i
For any event A, and partition cfw_B1,B2,.,B
STA257: Probability and
Statistics I
Section L0101 Fall 2015
Week 3, Lecture 6
Discrete Random Variables (2.1)
1
Discrete Probability
Distributions
List of common discrete probability
distributions:
Bernoulli
Binomial
Geometric
Negative Binomial
Hypergeom
Department of Mathematics, University of Toronto
MAT224H1S - Linear Algebra II
Winter 2017
Tutorial Problems 9
1. Textbook, Section 6.1, #1.
2.
Let S, T : V V be linear transformations such that ST = T S.
(a) Show that Im(S) and Ker(S) are invariant under
You are here
Assembly Language
Circuit Creation
Processors
Arithmetic
Logic Units
Devices
Finite State
Machines
Flip-flops
Circuits
Gates
Transistors
Making boolean expressions
So how would you represent boolean
expressions using logic gates?
Y = (A or B
STA257H1 F Term Test October 20, 2014
Section L0101
Last Name:_
First Name(s):_
Student #:_
TAs Name:_
Tutorial Room: _
Time allowed: 100 minutes.
Total marks = 50. Marks shown in brackets.
Check that you have all the consecutively numbered pages of th
STA25'7i-Il F Term Test October 20, 2014
Section L0101
Last Name: ,3 >
First Name(s): . O
\Q
Student #:
S ame: w
TA N 0%
Tutorial Room: %
Time allowedzwo minutes.
Total marks = 50. Marks shown in brackets.
Check that you have all the consecutively num
Last name,
First name:
. Student #:
ww
w.
ox
di
a
STA257H1 F, Section L0101, Term Test, October 21, 2013
Duration: 110 min. Allowed: hand-calculator and a formula sheet, one page two sided, or
two pages one sided, as posted on the website. Show your work
STA257
PROBABILITY
Ch 2, Random variables, Part 1
Instructor
Dragan Banjevic
Fall 2016
1
Ch. 2 Random Variables
Remember experiment: Observing weather today
Conditions
Outcomes temperature, pressure, precipitation,
wind, humidity, etc.
Events . , It w
STA257
PROBABILITY
Ch 2, Random variables, Part 2
Instructor
Dragan Banjevic
Fall 2016
1
Ch. 2 Continuous Variables
Remember experiment: Observing weather today
Outcomes temperature, pressure, precipitation,
wind, humidity, etc.
Events . , It will be ho
STA257
PROBABILITY
Ch 5, 6 Limit Theorems
Instructor
Dragan Banjevic
Fall 2016
1
The Law of Large Numbers (I)
We built our theory hoping to get a good model for real
phenomena of random experiment, random events, and their
probabilities. Our main interpre
STA256H5F
Final Exam Review
(Part IV Multivariate Probability
Distributions)
Part I Joint Distribution Function
Definition
Let
Y 1 and Y 2 be discrete random variables. The joint (or bivariate) probability mass
Y 1 and Y 2 is given by
function (pmf) for
p
STA256H5F Final Review Practice Problems Part I
1. Suppose a family contains two children of different ages, and we are interested in the gender of
these children. Let F denote that a child is female and M that the child is male and let a pair such
as F M
STA257H1 F Term Test October 22, 2014
Section L5101
Last Name:_
First Name(s):_
Student #:_
TAs Name:_
Tutorial Room: _
Time allowed: 100 minutes.
Total marks = 50. Marks shown in brackets.
Check that you have all the consecutively numbered pages of th
STA257H1 F Term Test October 28, 2015
Section L5101
Last Name:_
First Name(s):_
Student #:_
TAs Name:_
Tutorial Room: _
Time allowed: 100 minutes.
Total marks = 60. Marks shown in brackets.
Check that you have all the consecutively numbered pages of th
Stat 310 Homework 2 Key
Chapter 2, problems 2, 10, 11, 20, 23, 26, 40, 45, 47, 52. Due 9/16/99.
2.2. An experiment consists of throwing a fair coin 4 times. Find the frequency function
and the cumulative distribution function for the follwing random varia
Stat 310 Homework 1 Key
Chapter 1, problems 2, 18, 20, 22, 24, 48, 64, 72, 73, 75. Due 9/9/99.
1.2. Two six—sided dice are thrown sequentially, and the face values that come up are
recorded.
a) List the sample space. Well, noting the outcomes as (first die
Name:
ID:
Homework for 1/8
1. [4-6] Let X be a continuous random variable with probability density
function f (x) = 2x, 0 x 1.
(a) Find E[X].
(b) Find E[X 2 ].
(c) Find Var[X].
(a) We have
E[X] =
Z
1
xf (x) dx =
1
Z
1
x 2x dx =
0
2x3
3
1
=
0
2
.
3
(b) We
