Download Expectation Value Expectation Value Example Uniform Distribution

7/23/2014
Expectation Value
Online Review Course of
Undergraduate Probability and Statistics
• Discrete random variables are characterized by
their PMF (probability mass function)
Review Lecture 8
Expectation Value and Variance
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Chris A. Mack
• We define the Expectation Value (mean) of the
random variable X as
Adjunct Associate Professor
Course Website: www.lithoguru.com/scientist/statistics/review.html
© Chris Mack, 2014
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© Chris Mack, 2014
Expectation Value Example
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© Chris Mack, 2014
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Variance
• Consider a coin toss that produces a head
(success) with probability p
• Discrete random variables are characterized by
their PMF (probability mass function)
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© Chris Mack, 2014
Bernoulli Random Variable
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• The expectation value for a uniformly distributed
random variable is the arithmetic mean of the
possible values
+…
© Chris Mack, 2014
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• Let X = uniformly distributed, N discrete values
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Uniform Distribution
• Let X = sum of two six-sided, fair dice
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• We define the Variance of the random variable X
as
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© Chris Mack, 2014
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7/23/2014
Bernoulli Random Variable
Binomial Distribution
• Consider a coin toss that produces a head
(success) with probability p
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• Repeat a Bernoulli trial n times (p = probability of
success)
• Let X = number of successes
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Poisson Distribution
• How many coin tosses are required before the
first heads (success) comes up?
• Let X = number of tosses to get the first head
• Consider the Binomial distribution when n >> k
• More specifically, let n → ∞ while keeping
np = λ = constant
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© Chris Mack, 2014
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Independence
• For all cases,
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© Chris Mack, 2014
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• What is the “expectation value” and how is
it calculated for a discrete RV?
• What is the “variance” and how is it
calculated for a discrete RV?
• What is the mean and variance of the
Bernoulli RV and a binomial distributed RV?
• What is the variance of the sum of two
independent random variables?
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© Chris Mack, 2014
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Review #8: What have we learned?
:
• If X and Y are independent random variables
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A result of
independent
trials
© Chris Mack, 2014
Geometric Distribution
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© Chris Mack, 2014
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© Chris Mack, 2014
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