Vina Nguyen
HSSP – August 3, 2008
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Random Variables Intro (see last lecture)
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X = number of tosses needed for a head to
come up the first time
ƒ p = probability that a head is tossed
ƒ
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px(k) = (1-p)k-1p
T
k = [1,2,…]
T
T
k=4
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H
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PMF of the geometric RV
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Applications
ƒ Passing a test in a given try
ƒ Finding a missing item in a given search
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px(k) =
e-λ
λk
k!
k = 0,1,2…
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Good approximation for binomial RV if
ƒ λ = np
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[ n is large, p is small ]
Example: # of typos in a book with
ƒ n = total words
ƒ p = small probability of words being misspelled
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Example: # of cars involved in an accident on
a given day
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If n = 100, p=0.01
Result using the binomial PMF:
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Result using the Poisson PMF:
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Poisson RV is easier to calculate
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You go to a party with 500 guests. What is
the probability that exactly one other guest
has the same birthday as you? Calculate this
exactly and also approximately using the
Poisson PMF. (Exclude birthdays on February
29.)
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Bernoulli
Binomial
Geometric
Poisson
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Expectation
ƒ Mean
ƒ Expected Value
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Variance
ƒ Measure of how spread out a distribution is
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Consider 2 independent coin tosses, each
with a ¾ probability of a head, and let X be
the # of heads obtained.
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This is a binomial random variable with
parameters n=2, p=3/4
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What is the PMF?
What is its mean?
E[X] = ?
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E[X] = ∑ x px(x)
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Sum of all possible values of x multiplied by
their probabilities
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Var(X) = expected value of (X-E[X])2
ƒ E [ (X – E[X])2 ]
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Related quantity: Standard Deviation
ƒ StdDev(X) = √ Var(X)
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MIT OpenCourseWare
http://ocw.mit.edu
Probability: Random Isn't So Random
Summer 2008
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Vina Nguyen