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Review of Probability

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Definitions (1)

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Quiz 1.Let’s say I have a random variable X for a coin, with event space {H, T}. If the probability P(X=H) is 0.5, what is P(X=T)? 2. If P(X=H)=0.25, what is P(X=T)?

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Definitions (2) Joint distribution: A probability distribution of two or more random variables. The event space for this distribution is the cross product of the event space of the individual random variables. E.g. Let X 1 be a random variable for a coin flip. Let X 2 be a random variable for a second coin flip. P(X 1, X 2 ) is a joint distribution over all possible values for both coin flips.

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Quiz How many events are in the event space for flipping two coins? Name two of these events.

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Definitions (3) Marginal distribution: This is just any probability distribution, but people use it to refer to a distribution over one variable when they’ve separately introduced a joint distribution over that variable and a second variable. E.g., if I have a joint distribution P(X 1, X 2 ), then P(X 1 ) is a marginal distribution over X 1, and P(X 2 ) is a marginal distribution over X 2.

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Important Equation

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Definitions (4)

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Important Equation (2)

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Definitions (5)

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Quiz 1.Suppose I flip a coin 3 times. Each time has P(H)=0.5. Assume the three coin tosses are independent. What is P(H, H, H)? 2.Suppose I flip the coin 4 times, and let the random variable for the i-th time be Xi. What is P(X1=X2=X3=X4)? 3.What is the probability that, in 4 coin flips, I get at least 3 heads?

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Quiz Suppose I have one normal coin (P(X1=H)=0.5), and one weird coin with the following properties: P(X2=H|X1=H)=0.9 P(X2=T|X1=T)=0.8 If I flip X1 and then X2, what is P(X2=H)?

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Important Equations so far

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Quiz: Diabetes P(D)=0.01 (called the prior probability) Test for diabetes is either + or – P(+|D)=0.9 P(-| D)=0.8 P(-|D) = P(+| D) = P(+, D) = P(-, D) = P(+, D) = P(-, D) = P(D|+) =

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Bayes Rule Likelihood Prior Marginal Likelihood Posterior

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Bayes Rule for Diabetes Test P(D|+) = ?

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