Chapter 2.4 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Agenda Expected Value

# Chapter 2.4 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University

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Agenda Expected Value Variance and standard deviation Quantiles and percentiles
Expected value: discrete random variable Let ? be the number that comes up on a fair, six-sided die. If we roll it infinitely many times, on average, what number do you expect to get?

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Expected value: discrete random variable Definition: The expected value (expectation or mean) of a discrete random variable ? with p.m.f. ?(𝑥) is 𝜇 = ? ? = 𝑥?(𝑥) 𝑥 Represents weighted average of possible values of ? can take, each value being weighted by its probability to occur. Note: the random variable might never actually take on the ‘expected’ value You can’t roll a 7 2 on an ordinary die!
A lemma Let 𝐼 𝐴 is the indicator random variable for event 𝐴 , then ? 𝐼 𝐴 = 𝑃(𝐴) .

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Example: computer crash time A computer program crashes at the end of each hour of use with probability 𝑝 , if it has not crashed already. What is the expected time until

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