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CSE 531: Performance Analysis of SystemsLecture 2: Probs & Stats review
Anshul Gandhi1307, CS building
[email protected]@stonybrook.edu
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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.
• Law of total probability• Conditional probability• Bayes’ theorem
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Random variables• Discrete and Continuous
• Discrete Countable possibilities pmf
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Discrete RVs• PMF for sample space S
Pr[X = s] = pX(s) = p(s)
CDF: FX(a) = Pr[X ≤ a] =
Inverse CDF: F@X(a) = Pr[X > a] = 1 - FX(a) =
Mean E[X] =
E[X2] =
Var[X] = E[X2] – (E[X])2
Pr[ ]x a
X a
Pr[ ]x a
X a
. ( )s Ss p s
( ) 1s Sp s
2. ( )s Ss p s
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Bernoulli(p)• Outcome of a coin toss• p(1) = p• p(0) = 1-p
(find limits of s)
Mean E[X]
E[X2]
Var[X]
( ) 1s Sp s
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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)
Mean E[X]
E[X2]
Var[X]
( ) 1s Sp s
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Geometric(p)• Number of flips till we get a 1• p(i) = (1-p)(i-1) . p
Mean E[X]
E[X2]
Var[X]
( ) 1s Sp s
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Continuous RVs• PDF for sample space S
Pr[a ≤ X ≤ b] =
CDF: FX(a) = Pr[X ≤ a] =
E[Xi] =
Var[X] = E[X2] – (E[X])2
( ) ( )b b
Xa a
f x dx f x dx
( ) 1, ( ) Pr[ ]f x dx f x dx x X x dx
( )a
f x dx
( )a
ix f x dx
( ) ( )ddxf x F x
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Uniform(a, b)• f(x) = 1/(b-a) for a < x < b
E[X]
E[X2]
Var[X]
( ) 1f x dx
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Exponential(λ)• f(x) = λ e - λ x, x ≥ 0
E[X]
E[X2]
Var[X]
( ) 1f x dx
