ELEC 2600 Lecture 2: Build a Probability Model
Specifying Random Experiments
Sample spaces and events
Set Operations
The Three Axioms of Probability
Corollaries
Probability Laws for Assigning Probabilities
Elec2600 Lecture 2
1
Specifying Random Expe
Elec2600: Lecture 20
Sums of Random Variables
Mean and Variance of Sample Means
Useful Inequalities
Laws of Large Numbers
Elec2600 Lecture 20
1
Sums of Random Variables
For any set of random variables, X 1 , X 2 ,., X n
E
VAR
j
X j
Xj
j 1
n
E[ X
j
Elec2600: Lecture 14
Pairs of continuous random variables
Review of 2D functions, differentiation and integration
Joint cumulative distribution function
Joint density function
Elec2600 Lecture 14
1
Two Random Variables
One random variable can be cons
Elec 2600: Lecture 9
Important discrete random variables
Summary of variables you know:
Bernoulli
Binomial
Geometric
Discrete Uniform
New random variable: Poisson
*MATLAB commands for plotting probability mass functions
and generating discrete random
Elec2600 Lecture 12
Functions (or Transformations) of a Random Variable
Elec2600 Lecture 12
1
Functions/Transforms of a Random Variable
Problem statement:
Given a random variable X with known distribution and a real valued
function g(x), such that Y = g(
Elec2600: Lecture 22
Definition of a Random Process
Specification of a Random Process
Elec2600 Lecture 22
1
Definition of a Random Process
Definition: A random process or
stochastic process maps a
probability space S to a set of
functions, X(t,)
It assi
Elec2600: Lecture 24
Discrete Time Random Processes
Sum Processes
ISI Processes
Elec2600 Lecture 24
1
Sum Random Processes
Definition: A sum process Sn is obtained by taking the sum of
all past values of an i.i.d. random process Xn, i.e.,
n
S n X i X
Elec2600: Lecture 15
Independence
Expectation of Function of 2 Variables
Joint Moments
Elec2600 Lecture 15
1
Independence
Definition: Two random variables X and Y are said to be
independent or statistically independent if for any events, AX
and AY, de
Elec 2600: Lecture 3
Computing Probabilities using Counting Methods
Sample Size Computation and Examples
Probabilities and Poker!
Elec2600 Lecture 3
1
Computing Probabilities Using Counting Methods
In experiments where the outcomes are equiprobable, w
Elec2600: Lecture 26
Stationary Random Processes
Wide Sense Stationary (WSS) Random Processes
Elec2600 Lecture 26
1
Stationary Random Processes
Definition: A process is stationary if the joint distribution of any set of samples does not
depend on the pl
Elec2600: Lecture 16
Conditional Probability
Conditional Expectation
Elec2600 Lecture 16
1
Conditional Probability Mass Functions
Suppose that X and Y are discrete RVs assuming integer values.
The conditional pmf of Y given X is
pY | X (k | j )
P Y k
a.) z
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ELEC 210 Fall09 Solutions to HW8
1. Suppose that X and Y are discrete random variables. Assume that X is uniform on (1,2,3,4,5)
and that Y is chosen at random between 1 and X.
a) Write the conditional pmf of Y given X, pY | X (k | j ) for j (1, 2,3, 4,5)
ELEC210 Fall09
Solutions for HW6
1. Solutions
(a) The event min(X, Y) -2 is equivalent to X -2 and Y -2,
similarly, the event maxcfw_X, Y 8 is equivalent to the event X 8
and Y 8. So the event cfw_min(X, Y) -2cfw_max(X, Y) 8 is
equivalent to -2 X 8 and -2
ELEC210 Fall09
Questions of HW 7
1. Suppose that X and Y are independent random variables where X is
uniformly distributed in [0, 1] and Y is exponentially distributed with
parameter = 0.5. Find
a) P[cfw_X 0.75cfw_|Y| 3]
b) P[Y < X]
2. Suppose that X and
ELEC 210 Fall09
Questions of HW 8
1. Suppose that X and Y are discrete random variables. Assume that X is uniform on (1,2,3,4,5)
and that Y is chosen at random between 1 and X.
a) Write the conditional pmf of Y given X, PY | X (k | j ) for j (1, 2,3, 4,5)
ELEC210 Fall09
Questions of HW 6
1. Let X and Y be continuous random variables that have joint cumulative
distribution given by FX,Y(x,y). Express the probability of the following
events in terms of FX,Y(x,y).
a) cfw_min(X, Y) -2cfw_max(X, Y) 8
b) cfw_X 3
Elec2600: Lecture 17
One function of two random variables
Discrete random variables
Continuous random variables
Using conditioning
Thus far, for Z = g(X,Y), with X and
Y random variables, we know how to
compute the moments of Z.
But how do we compute
Elec2600 Lecture 11
Expectation of Continuous Random Variables
Variance of Continuous Random Variables
Important Continuous Random Variables
Elec2600 Lecture 11
1
Review: Expectation
Interpretation
The average value of a random variable if we repeat
Elec 2600: Lecture 5
Sequential Experiments
Bernoulli trials
Binomial probability law
*Multinomial probability law
Geometric probability law
Sequences of dependent experiments
Example: Bean Machine Game!
Elec2600 Lecture 5
1
Sequential Experiments
Elec 2600: Lecture 8
Conditional Probability Mass Function
Conditional Expected Value
Conditional Poker Hand
Elec2600 Lecture 8
1
Conditional Probability Mass Function
The effect of partial information about the outcome of a random
experiment on the p
Elec2600: Lecture 22
Definition of a Random Process
Specification of a Random Process
Elec2600 Lecture 22
1
Definition of a Random Process
Definition: A random process or
stochastic process maps a
probability space S to a set of
functions, X(t,)
It assi
Elec2600: Lecture 18
Random Vectors
Joint distribution/density/mass functions
Marginal statistics
Conditional densities
Independence and Expectation
http:/www.cs.princeton.edu/~cdecoro/eigenfaces/
Elec2600 Lecture 18
1
N Random Variables
An N dimensional
Elec2600: Lecture 20
Sums of Random Variables
Mean and Variance of Sample Means
Useful Inequalities
Laws of Large Numbers
Elec2600 Lecture 20
1
Sums of Random Variables
For any set of random variables, X 1 , X 2 ,., X n
E
VAR
j
X j
Xj
j 1
n
E[ X
j]
j
n
Elec2600: Lecture 14
Pairs of continuous random variables
Review of 2D functions, differentiation and integration
Joint cumulative distribution function
Joint density function
Elec2600 Lecture 14
1
Two Random Variables
One random variable can be consid
Elec2600: Lecture 15
Independence
Expectation of Function of 2 Variables
Joint Moments
Elec2600 Lecture 15
1
Independence
Definition: Two random variables X and Y are said to be
independent or statistically independent if for any events, AX
and AY, de
Elec 2600: Lecture 4
Conditional Probability
Properties
Thus far, we have looked at the probability
of events occurring individually, without
regards to any other event.
However, if we KNOW that a particular
event occurs, then how does this change the
p
Elec 2600: Lecture 5
Sequential Experiments
Bernoulli trials
Binomial probability law
*Multinomial probability law
Geometric probability law
Sequences of dependent experiments
Example: Bean Machine Game!
Elec2600 Lecture 5
1
Sequential Experiments
Elec 2600: Lecture 3
Computing Probabilities using Counting Methods
Sample Size Computation and Examples
Probabilities and Poker!
Elec2600 Lecture 3
1
Computing Probabilities Using Counting Methods
In experiments where the outcomes are equiprobable, we ca
Elec2600: Lecture 25
Continuous Time I.S.I. Random Processes
Poisson Random Process
Additional Random Processes (FYI)
Random Telegraph Process
Shot Noise Process
Weiner Process
Elec2600 Lecture 25
1
The Poisson Process
Consider the following sequen