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# lecture10 - COMPSCI 102 Introduction to Discrete...

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COMPSCI 102 Introduction to Discrete Mathematics

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CPS 102 Classics
Today, we will learn about a formidable tool in probability that will allow us to solve problems that seem really really messy…

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If I randomly put 100 letters into 100 addressed envelopes, on average how many letters will end up in their correct envelopes? = k k (…aargh!!…) Hmm… k k Pr(k letters end up in correct envelopes)
On average, in class of size m, how many pairs of people will have the same birthday? k k Pr(exactly k collisions) = k k (…aargh!!!!…)

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The new tool is called “Linearity of Expectation”
Random Variable To use this new tool, we will also need to understand the concept of a Random Variable Today’s lecture: not too much material, but need to understand it well

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Random Variable A Random Variable is a real-valued function on S Examples: X = value of white die in a two-dice roll X(3,4) = 3, X(1,6) = 1 Y = sum of values of the two dice Y(3,4) = 7, Y(1,6) = 7 W = (value of white die) value of black die W(3,4) = 3 4 , Y(1,6) = 1 6 Let S be a sample space in a probability distribution
Tossing a Fair Coin n Times S = all sequences of {H, T} n D = uniform distribution on S D(x) = (½) n for all x S Random Variables (say n = 10) X = # of heads X(HHHTTHTHTT) = 5 Y = (1 if #heads = #tails, 0 otherwise) Y(HHHTTHTHTT) = 1, Y(THHHHTTTTT) = 0

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Notational Conventions Use letters like A, B, E for events Use letters like X, Y, f, g for R.V.’s R.V. = random variable
Two Views of Random Variables Input to the function is random Randomness is “pushed” to the values of the function Think of a R.V. as A function from S to the reals Or think of the induced distribution on

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0 1 2 TT HT TH HH ¼ ¼ ¼ ¼ S Two Coins Tossed X: {TT, TH, HT, HH} → {0, 1, 2} counts the number of heads ¼ ½ ¼ Distribution on the reals
It’s a Floor Wax And a Dessert Topping It’s a function on the sample space S It’s a variable with a probability distribution on its values You should be comfortable with both views

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lecture10 - COMPSCI 102 Introduction to Discrete...

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