Lecture Stat 302 Introduction to Probability - Slides 1
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Administrative details
Arnaud Doucet, O ce LSK 308c, Department of Statistics & O ce ICCS 189, Department of Computer Science. O ce Hour, Department of Statistics:
Lecture Stat 302 Introduction to Probability - Slides 18
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Jointly Distributed Random Variables
If both X and Y are continuous r.v., then their joint p.d.f. is a non-negative function f (x , y ) such that for any set C
Lecture Stat 302 Introduction to Probability - Slides 19
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Sum of Independent Random Variables
Consider Z = X + Y where X and Y are disrete r.v. of respective p.m.f. pX (x ) and pY (y ) then pZ ( z ) =
pX ( z
y
y ) pY
Lecture Stat 302 Introduction to Probability - Slides 20
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Conditional Distributions: Discrete Case
Given a joint p.m.f. for two r.v. X , Y it is possible to compute the conditional p.m.f. X given Y = y . Assume X , Y a
Lecture Stat 302 Introduction to Probability - Slides 21
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Conditional Distributions: Discrete Case
Given two r.v. X , Y , we have Discrete p (x , y ) p ( x ,y ) pX j Y ( x j y ) = p ( y ) Y E ( g (X )j y ) = g (x ) .pX
Lecture Stat 302 Introduction to Probability - Slides 22
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Characterizing Joint Distributions/Densities: Covariance
Consider two r.v. X and Y (either discrete or continuous), then the covariance of (X , Y ) is dened as
Lecture Stat 302 Introduction to Probability - Slides 23
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Exercise 1
Alf and Beth are two UBC students. They must take 4 300-level courses from a list of 12 possibilities. If they select their courses independently and
1. (a) E (b) A
2. B
3. (a) C (b) A (c) A (d) C
4. D
5.
1
Practice Final A
0.5
yaxis
1.0
Question 5 Part I
The area we are interested in is enclosed in the gure below, where the light gray shade represents
the square [0, 1][0, 1], and the darker area repre
STAT302: Introduction to Probability
Fall 2014-15
Assignment 3
Due date: see course website (Under Schedule).
Instructions
Academic integrity policy: I encourage you to discuss verbally with other students about the assignment.
However, you should write y
STAT302: Introduction to Probability
Fall 2014-15
Assignment 4
Due date: see course website (Under Schedule).
Instructions
Academic integrity policy: I encourage you to discuss verbally with other students about the assignment.
However, you should write y
STAT302: Introduction to Probability
Fall 2014-15
Assignment 3
Due date: see course website (Under Schedule).
Instructions
Academic integrity policy: I encourage you to discuss verbally with other students about the assignment.
However, you should write y
STAT302: Introduction to Probability
Fall 2014-15
Assignment 1
Due date: see course website (Under Schedule).
Instructions
Academic integrity policy: I encourage you to discuss verbally with other students about the assignment.
However, you should write y
STAT302: Introduction to Probability
Fall 2014-15
Assignment 2
Due date: see course website (Under Schedule).
Instructions
Academic integrity policy: I encourage you to discuss verbally with other students about the assignment.
However, you should write y
Lecture Stat 302 Introduction to Probability - Slides 17
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Jointly Distributed Random Variables
Assume we have two r.v. X and Y , then we dene the joint c.d.f. F (a, b ) = P (X The c.d.f of X is FX (a) = P (X a, Y ) a,
Lecture Stat 302 Introduction to Probability - Slides 16
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Change of Variables
Let X be a r.v. of pdf fX (x ) and consider the r.v. Y = g (X ) . A legitimate question is to ask what is the pdf fY (y ) of Y . This has numer
Lecture Stat 302 Introduction to Probability - Slides 15
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Continuous Random Variable
Let X a (real-valued) continuous r.v. It is characterized by its pdf f : R ! [0, ) which such that for any set A of real numbers P (X
Lecture Stat 302 Introduction to Probability - Slides 2
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Recapitulation
Principle of counting: If experiment 1 has n1 possible outcomes, experiment 2 has n2 possible outcomes,., experiment r nr possible outcomes, then th
Lecture Stat 302 Introduction to Probability - Slides 3
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Sample Space
Denition. The sample space S of an experiment (whose outcome is uncertain) is the set of all possible outcomes of the experiment. Example (child): Determ
Lecture Stat 302 Introduction to Probability - Slides 4
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Axioms of Probability
Consider an experiment with sample space S . For each event E , we assume that a number P (E ), the probability of the event E , is dened and
Lecture Stat 302 Introduction to Probability - Slides 5
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Conditional Probabilities
Conditional Probability. Consider an experiment with sample space S . Let E and F be two events, then the conditional probability of E gi
Lecture Stat 302 Introduction to Probability - Slides 6
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Independence
We say that the events fEi gn=1 are independent if i P (\n=1 Ei ) = i
i =1 n
P ( Ei ) .
and if for i1 , i2 , ., ir where ij 6= ik for j , k 2 f1, .,
Lecture Stat 302 Introduction to Probability - Slides 7
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Simpson Paradox: Sex Bias in Graduate Admissions? s
The University of California at Berkeley was sued for bias against women who had applied for admission to gradu
Lecture Stat 302 Introduction to Probability - Slides 8
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Expected Value
The expectation of X , denoted E (X ), is dened as E (X ) =
i =1
xi p (xi ) =
x p (x )
x :p (x ) >0
It corresponds to a weighted average of the po
Lecture Stat 302 Introduction to Probability - Slides 9
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Example: Optimal Stock
Vancouver 2010 Olympic Torchbearer red mittens are currently sold at a net prot of b $ for each unit sold and will bring a net loss of l $ f
Lecture Stat 302 Introduction to Probability - Slides 10
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Discrete Random Variables
A discrete r.v. X takes at most a countable number of possible values fx1 , x2 , .g with p.m.f. p (xi ) = P (X = xi ) where p (xi ) Expe
Lecture Stat 302 Introduction to Probability - Slides 11
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Discrete Random Variables
A discrete r.v. X takes at most a countable number of possible values fx1 , x2 , .g with p.m.f. p (xi ) = P (X = xi ) where p (xi ) Ex
Lecture Stat 302 Introduction to Probability - Slides 12
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Hypergeometric Random Variable
Consider a barrel or urn containing N balls of which m are white and N m are black. We take a simple random sample (i.e. without
Lecture Stat 302 Introduction to Probability - Slides 13
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Continuous Random Variables
For the time being, we have only considered discrete random variables (r.v.) - set of possible values is nite or countable - such as
Lecture Stat 302 Introduction to Probability - Slides 14
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Continuous Random Variable
Formaldenition: We say that X is a (real-valued) continuous r.v. if there exists a nonnegative function f : R ! [0, ) such that for