Transcript ppt
Review of Probability Theory
[Source: Stanford University]
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Random Variable
A random experiment with set of outcomes
Random variable is a function from set of
outcomes to real numbers
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Example
Indicator random variable:
A : A subset of
is called an event
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CDF and PDF
Discrete random variable:
The possible values are discrete (countable)
Continuous random variable:
The rv can take a range of values in R
Cumulative Distribution Function (CDF):
PDF and PMF:
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Expectation and higher
moments
Expectation (mean):
if X>0 :
Variance:
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Two or more random variables
Joint CDF:
Covariance:
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Independence
For two events A and B:
Two random variables
IID : Independent and Identically Distributed
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Useful Distributions
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Bernoulli Distribution
The same as indicator rv:
IID Bernoulli rvs (e.g. sequence of coin
flips)
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Binomial Distribution
Repeated Trials:
Number of times an event A happens among n
trials has Binomial distribution
Repeat the same random experiment n times. (Experiments are
independent of each other)
(e.g., number of heads in n coin tosses, number of arrivals in n
time slots,…)
Binomial is sum of n IID Bernoulli rvs
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Mean of Binomial
Note that:
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Binomial - Example
0.45
n=4
0.4
0.35
p=0.2
n=10
0.3
0.25
n=20
0.2
n=40
0.15
0.1
0.05
0
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Binomial – Example (ball-bin)
There are B bins, n balls are randomly
dropped into bins.
: Probability that a ball goes to bin i
: Number of balls in bin i after n drops
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Multinomial Distribution
Generalization of Binomial
Repeated Trails (we are interested in more
than just one event A)
A partition of W into A1,A2,…,Al
Xi shows the number of times
among n trials.
Ai occurs
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Poisson Distribution
Used to model number of arrivals
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Poisson Graphs
0.5
l=.5
0.45
0.4
l=1
0.35
0.3
0.25
l=4
0.2
l=10
0.15
0.1
0.05
0
0
5
10
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Poisson as limit of Binomial
Poisson is the limit of Binomial(n,p) as
Let
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Poisson and Binomial
0.4
n=5,p=4/5
0.35
Poisson(4)
0.3
0.25
n=10,p=.4
0.2
n=20, p=.2
0.15
0.1
n=50,p=.08
0.05
0
0
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Geometric Distribution
Repeated Trials: Number of trials till some
event occurs
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Exponential Distribution
Continuous random variable
Models lifetime, inter-arrivals,…
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Minimum of Independent
Exponential rvs
: Independent Exponentials
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Memoryless property
True for Geometric and Exponential Dist.:
The coin does not remember that it came up tails l times
Root cause of Markov Property.
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Proof for Geometric
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Characteristic Function
Moment Generating Function (MGF)
For continuous rvs (similar to Laplace
transform)
For Discrete rvs (similar to Z-transform):
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Characteristic Function
Can be used to compute mean or higher
moments:
If X and Y are independent and T=X+Y
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Useful CFs
Bernoulli(p) :
Binomial(n,p) :
Multinomial:
Poisson:
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