Some Rules of Probability

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Some Rules of Probability
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CONDITIONAL PROBABILITY

• More formulae
• P(BA)
• Thus, P(BA) is not the same as P(AB).
• P(A?B) P(AB)P(B)
• P(A?B) P(BA)P(A)

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• AIDS Testing Example
• ELISA test
• HIV positive
• HIV negative
• Correctness 99 on HIV positive person
• (1 false negative)
• 95 on HIV negative person
• (5 false alarm)
• Mandatory ELISA testing for people applying for
  marriage licenses in MA.
• low risk population 1 in 500 HIV positive
• Suppose a person got ELISA .
• Q HIV positive?

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Bayes Theorem
By the way This is the key result
underlying Bayesian Statistics
some people make a living out of this formula
Try Michael Birnbaums (former UIUC psych
faculty) Bayesian calculator http//psych.fullerto
n.edu/mbirnbaum/bayes/BayesCalc.htm
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Bayes Theorem
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Today

• The Binomial Distribution

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We have already talked about the most famous
continuous random variable, which, because it
is so heavily used, even has a name The Normal
Random Variable (and, associated with it, the
Normal Distribution)
Today we will talk about a famous discrete random
variable, which, because it is so heavily used,
also has a name The Binomial Random Variable
(and, associated with it, the Binomial
Distribution)
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FAIR COIN POPULATION THEORETICAL PROBABILITY
OF HEADS IS ½
If you toss a fair coin 10 times, what is the
probability of x many heads?
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Binomial Random Variable
Potential Examples Repeat a simple dichotomous
experiment n times and count Coin Tossing
heads Marketing purchases Medical
procedure patients cured Finance days
stock ? Politics voters favoring a given
bill Testing number of test items of a given
difficulty that you solve
correctly Sampling of females in a random
sample of people Education of high school
students who drink alcohol
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Population
Repeat simple experiment n many times
independently.
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Binomial Random Variable
X number of successes in n many
independent repetitions of an experiment, each
repetition having a probability p of success
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The Binomial Distribution
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Trial 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
Trial 2 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0
Trial 3 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
Trial 4 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
Probab. (letting q1-p) pppp pppq ppqp ppqq pqpp pqpq pqqp pqqq qppp qppq qpqp qpqq qqpp qqpq qqqp qqqq
X 4 3 3 2 3 2 2 1 3 2 2 1 2 1 1 0
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Probab. (letting q1-p) pppp pppq ppqp ppqq pqpp pqpq pqqp pqqq qppp qppq qpqp qpqq qqpp qqpq qqqp qqqq
X 4 3 3 2 3 2 2 1 3 2 2 1 2 1 1 0
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In general, for n many trials
How do we figure that out in general?
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Factorial
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Binomial Distribution Formula
TABLE C Pages T-6 to T-10 in the book
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Probab. (letting q1-p) pppp pppq ppqp ppqq pqpp pqpq pqqp pqqq qppp qppq qpqp qpqq qqpp qqpq qqqp qqqq
X 4 3 3 2 3 2 2 1 3 2 2 1 2 1 1 0
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BINOMIAL COEFFICIENTS
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Example heads in 5 tosses of a coin
XB(5,1/2) Expectation Variance
heads in 5 tosses of a coin 2.5
1.25
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PROOF
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Another Statistic The Sample Proportion
Remember that X is a random variable
The sample proportion is a linear transformation
of X and thus a random variable too
Sampling Distribution of the Sample Proportion
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The Sample Proportion
Unbiased Estimator
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Lets take another look at some Binomial
Distributionsespecially what happens as n gets
bigger and bigger
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Normal Approximation of/to the Binomial
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Normal Approximation
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Normal Approximation Lets check it out!
TABLE C Pages T-6 to T-10 in the book
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Normal Approximation Lets check it out!
Standardizing
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Normal Approximation Lets check it out!
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Normal Approximation Lets check it out!
Are we stuck with a bad approximation??
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For now, thats it

• We will revisit the Binomial
• Based on the sample proportion as an estimate of
  the population proportion, we will develop
  confidence intervals for the population
  proportion.
• We will carry out hypothesis tests about the true
  population proportion, using the information
  gained from the sample proportion.