Download Probs & Stats 1

CSE 531: Performance Analysis of Systems
Lecture 2: Probs & Stats review
Anshul Gandhi
1307, CS building
[email protected]
[email protected]
1
Outline
1. Announcements
2. Probability basics
 Experiments, events, helpful relations
3. Random variables
 Discrete

Bernoulli, Binomial, Geometric
 Continuous

Uniform, Exponential
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Announcements
• Collaborating on assignments
•
Assignment 1 (next week)
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Basics
•
•
•
Probability is defined in terms of some experiment.
The set of all outcomes of an experiment is its sample space.
A subset of the sample space is called an event.
 Mutually exclusive
 Partition
 Independent
•
A function defined on the outcomes is a random variable.
•
•
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Law of total probability
Conditional probability
Bayes’ theorem
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Random variables
•
Discrete and Continuous
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Discrete
 Countable possibilities
 pmf
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Discrete RVs
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PMF for sample space S
 Pr[X = s] = pX(s) = p(s)
  p( s)  1
sS
 CDF: FX(a) = Pr[X ≤ a] =  Pr[ X  a]
xa
 Inverse CDF: F̅X(a) = Pr[X > a] = 1 - FX(a) =  Pr[ X  a]
xa
 Mean E[X] =  s. p ( s )
sS

E[X2]
2
s
=  . p( s)
sS
 Var[X] = E[X2] – (E[X])2
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Bernoulli(p)
• Outcome of a coin toss
• p(1) = p
• p(0) = 1-p
  p( s)  1
sS
(find limits of s)
 Mean E[X]
 E[X2]
 Var[X]
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Binomial(n, p)
• Number of 1’s when flipping a Bernoulli coin n times
• p(i) = nCi pi (1-p)(n-i)
  p( s)  1
sS
 Mean E[X]
 E[X2]
 Var[X]
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Geometric(p)
• Number of flips till we get a 1
• p(i) = (1-p)(i-1) . p
  p( s)  1
sS
 Mean E[X]
 E[X2]
 Var[X]
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Continuous RVs
•
PDF for sample space S
b
b
a
a
 Pr[a ≤ X ≤ b] =  f X ( x)dx   f ( x)dx

  f ( x) dx  1, f ( x) dx  Pr[ x  X  x  dx]

a
 CDF: FX(a) = Pr[X ≤ a] =  f ( x ) dx

 f ( x) 
d
dx
F ( x)
a

E[Xi]
i
x
=  f ( x ) dx

 Var[X] = E[X2] – (E[X])2
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Uniform(a, b)
• f(x) = 1/(b-a) for a < x < b

  f ( x)dx  1

 E[X]
 E[X2]
 Var[X]
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Exponential(λ)
• f(x) = λ e - λ x, x ≥ 0

  f ( x)dx  1

 E[X]
 E[X2]
 Var[X]
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