Lecture 4
Continuous Random Variables
Continuous Probability Space
is not countable (an interval, a subset of )
Outcome
can be any real number
We cannot assign probabilities to each outcome
Example: = [0, 1]
Let
and
How to define ?
Idea:
should be weig
Lecture 7
Joint Distribution
Joint Distribution
Let , be random variables. Then we define their joint cumulative distribution
function to be
, ( , ) = ( , )
Similarly, for n-dimensional case:
1, ( 1 , , ) = (1 1 , , )
Discrete case:
Let , be random vari
Lecture 8
Back to Expected Value and
Variance
Back to Expected Value and Variance
Properties of Expected Value:
( + ) = ( ) + ( ), ,
If , are independent, then ( ) = ( ) ()
Covariance
For random variables , with ( ), ( ) < the covariance of and is
( ,
Lecture 9
Convolution. Multivariable
Transformations.
Convolution
Let , be jointly distributed random variables. Consider = + .
What is the probability/density function of ?
Discrete Case: Let , have joint probability mass function , (, ).
Note: = wheneve
Lecture 10
Order Statistics. Generating
Functions.
Order Statistics
Let 1 , , be i.i.d., i.e. with the same cdf, (), and df, ().
The order statistics of a set of random variables 1 , , are the same random variables
but arranged in increasing order.
Denote
Lecture 11
Convergence
Convergence in Distribution
Recall:
Definition: Let 1 , 2 , be a sequence of random variables with cdf's 1 , 2 , , and let
be a random variable with cdf (). We say that the sequence cfw_ converges in
distribution to if
lim () = ()
Midterm Review
What to do:
Read lectures 1-6
Do the assigned exercises from the textbook
Go over the quiz questions
Use sample test to practice
Use extra TAs' office hours
1. Given that A and B are independent with ( ) = 0.8 and ( ) = 0.3, find
P(A).
Lecture 3
Random Variables
Random Variables
A random variable is a variable whose value is a numerical outcome of a random
phenomenon, so its values are determined by chance. We shall use letters such as X or Y
to represent a random variable, and x or y t
Lecture 5
Expected Value. Variance.
Functions of Random Variables.
Expected Values (continued)
Theorem: Given
.
If X is a discrete random variable, then
[ ( )]
( )
( )
If X is a continuous random variable, then
[ ( )]
Example:
~ Poisson(),
Example:
~ Uni
Midterm Test Info
Date:
L0101: Monday, October 20, 3:10 - 4:50 pm (100 minutes)
L5101: Wednesday, October 22, 7:10 - 8:50 pm (100 minutes)
Location:
L0101:
L5101:
Last Name
Room
Last Name
Room
AV
EX100
AL
GB404
WZ
EX310
MZ
GB405
Content: Lectures 1-6
Don'
STA257H1 F - Probability and Statistics I - Fall 2014
Instructor: Olga Chilina
Office: SS6002
Office hours: Mondays, 2:00 - 3:00 pm and Wednesdays, 5:00 - 6:00 pm
Email: [email protected]
Webpage: http:/www.utstat.toronto.edu/~olgac/
Lecture time a