Utility
Utility is value of some choice
2 choices, each with n consequences: c1, c2,., cn
One of ci will occur with probability pi
Each consequence has some value (utility): U(ci)
Which choice do you make?
Example: Buy a $1 lottery ticket (for $1M prize)?
Discrete Joint Mass Functions
For two discrete random variables X and Y, the
Joint Probability Mass Function is:
p X ,Y (a, b) = P( X = a, Y = b)
Marginal distributions:
p X (a ) = P ( X = a ) = p X ,Y (a, y )
y
pY (b) = P(Y = b) = p X ,Y ( x, b)
x
Exampl
Balls, Urns, and the Supreme Court
Supreme Court case: Berghuis v. Smith
If a group is underrepresented in a jury pool, how do you tell?
Article by Erin Miller Friday, January 22, 2010
Thanks to (former CS109er) Josh Falk for this article
Justice Breyer
Binary Search Tree
A binary search tree (BST), is a binary tree where for
every node n in the tree:
n's value is greater than all the values in its left subtree.
n's value is less than all the values in its right subtree.
both n's left and right subtrees
CS 109
Spring 2015
Homework 4
May 1st, 2015
Due: Friday, May 15th, 2015 5:00 PM
For each problem, briefly explain/justify how you obtained your answer.
This doesnt need to be a paragraph or even complete sentences.
This can be as simple as writing out y
CS 109
Spring 2015
Homework 2
April 10th, 2015
Due: April 20th, 2015 5:00PM
For each problem, briefly explain/justify how you obtained your answer.
This doesnt need to be a paragraph or even complete sentences.
This can be as simple as writing out your
Welcome Back Our Friend: Expectation
Recall expectation for discrete random variable:
E[ X ] = x p ( x)
x
Analogously for a continuous random variable:
E[ X ] =
x f ( x) dx
Note: If X always between a and b then so is E[X]
More formally:
if P (a X b) = 1
Computing Probabilities from Data
Various probabilities you will need to compute for
Naive Bayesian Classifier (using MLE here):
# instances in class = 0
P (Y = 0) =
total # instances
# instances where X i = 0 and class = 0
P ( X i = 0, Y = 0) =
total # i
From Urns to Coupons
Coupon Collecting is classic probability problem
Ask questions like:
There exist N different types of coupons
Each is collected with some probability pi (1 i N)
After you collect m coupons, what is probability you
have k different kin
Normal Random Variable
X is a Normal Random Variable: X ~ N(, 2)
Probability Density Function (PDF):
1
( x ) 2 / 2 2
f ( x) =
e
2
E[ X ] =
Var ( X ) =
2
where < x <
f (x)
Also called Gaussian
Note: f(x) is symmetric about
Common for natural phenome
n
Recursive definition of
k
Lets write a function C(n, k)
The number of ways to select k objects from a set of n objects.
C(n,k)
C(n,k)
Select any one of the n points in the group
C(n,k)
Separate this point from the rest
C(n,4)
Lets consider specific pr
Introduction to Probability
CS109 Lecture 3
Elmer Le & Kevin Shin
June 24, 2015
Le, Shin
Intro Probability
Outline
1
Set theoretic Background
Notation
Sample Space
Event Space
Set Operations on Events
2
Axioms of Probability
Axioms
Implications of Axioms
Introduction to Probability for Computer Scientists
CS109 Lecture 6
Kevin Shin
July 1, 2015
Le, Shin
Cond Indep, Random Variables, Mass Functions, & Expectation
Outline
1
What is Conditional Independence?
Conditional Independence Intuition
Conditional Ind
Introduction to Probability for Computer Scientists
CS109 Lecture 2
Elmer Le & Kevin Shin
June 25, 2015
Le, Shin
Counting: Combinatorics
Outline
1
Why Learn about More Counting?
2
Counting Ordered Arrangements of Distinct Objects: Permutations
Permutation
Introduction to Probability for Computer Scientists
CS109 Lecture 1
Elmer Le & Kevin Shin
June 22, 2015
Le, Shin
Motivation & Counting
Outline
1
Introduction
Denition of Probability
Topics in Probability
2
Motivation
Probability in Computer Science
Probab
Not Everything is Equally Likely
Say n balls are placed in m urns
Each ball is equally likely to be placed in any urn
Counts of balls in urns are not equally likely!
Example: two balls (A and B) placed with equal
likelihood in two urns (Urn 1 and Urn 2)
P
Whither the Binomial
Recall example of sending bit string over network
n = 4 bits sent over network where each bit had
independent probability of corruption p = 0.1
X = number of bits corrupted. X ~ Bin(4, 0.1)
In real networks, send large bit strings (le
Sample Spaces
Sample space, S, is set of all possible outcomes
of an experiment
Coin flip:
Flipping two coins:
Roll of 6-sided die:
# emails in a day:
YouTube hrs. in day:
S = cfw_Head, Tails
S = cfw_(H, H), (H, T), (T, H), (T, T)
S = cfw_1, 2, 3, 4, 5, 6
The Tragedy of Conditional Probability
Thanks xkcd!
http:/xkcd.com/795/
A Few Useful Formulas
For any events A and B:
P(A B)
= P(B A)
(Commutativity)
P(A B)
= P(A | B) P(B)
= P(B | A) P(A)
(Chain rule)
P(A Bc) = P(A) P(AB)
P(A B)
P(A) + P(B) 1
(Intersect
What is Machine Learning?
Many different forms of Machine Learning
We focus on the problem of prediction
Want to make a prediction based on observations
Vector X of m observed variables: <X1, X2, , Xm>
o
o
Based on observed X, want to predict unseen varia
Inequality, Probability, and Joviality
In many cases, we dont know the true form of a
probability distribution
E.g., Midterm scores
But, we know the mean
May also have other measures/properties
o
Variance
o
Non-negativity
o
Etc.
Inequalities and bounds st
Great (Conditional) Expectations
X and Y are jointly discrete random variables
Recall, conditional expectation of X given Y = y:
E[ X | Y = y ] = x P ( X = x | Y = y ) = x p X |Y ( x | y )
x
x
Analogously, jointly continuous random variables:
E[ X | Y = y
Probability and Random Variables / Processes for Wireless Communications
Professor Aditya K Jagannathan
Department of Electrical Engineering
Indian Institute of Technology, Kanpur
Module Number 03
Lecture Number 14
Gaussian Random Variable and Linear Tran
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 2
Lecture 12
Application: Average Delay and RMS Delay Spread of Wir
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module 2
Lecture No 7
Bayes Theorem and Aposteriori Probabitiesr
Hello, welcom
Probability and Random Variables/Processes for Wireless Communications.
Professor Aditya K Jagannatham.
Department of Electrical Engineering.
Indian Institute of Technology Kanpur.
Lecture -16.
Application: array processing and Array Gain with uniform lin
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 2
Lecture 8
Maximum Aposteriori Probability (MAP) Receiver
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Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 1
Lecture 2
Axioms of probability
Hello, welcome to to another modu
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 2
Lecture 10
Applications: Power of Fading Wireless Channel
Hello,
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 2
Lecture 9
Random Variables, Probability Density Function (PDF)
He
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 1
Lecture 1
Basics -Sample Space and Events
Hello, welcome to this
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module 4
Lecture No 23
Gaussian Process Through LTI Syatem Example:WGN Through
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 4
Lecture 21
Transmission of WSS Random Process Through LTI System
Probability and Random Variables/Processes for Wireless Communication
Professor Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology Kanpur
Module No. 1
Lecture 6
Independent Events-Multiantenna FadingExample.
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