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Lecture 03

# Lecture 03 - 1 Lecture Plan • Finite Sample Space(Review...

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Unformatted text preview: 1 Lecture Plan • Finite Sample Space (Review). • Conditional Probability • Independence • Random Variables • Expected Values • Markov’s Inequality 2 Finite Sample Space (Review) S = { w 1 ,w 2 ,...,w N } and let p i = P ( { w i } ) , i = 1 ,...,N. By axiom A1 we have p i ≥ 0 by axiom A2 and A3 applied to { w 1 } ,..., { w N } , we have ∑ N i =1 p 1 = 1 . From A3 for any subset A of S we have P ( A ) = ∑ i : w i ∈ A p i . What if all the outcomes are equally likely? Then p i = p for every i = 1 ,...,N and 1 = ∑ N i =1 p i = Np so p = 1 /N, and P ( A ) is equal to the number of elements of A divided by N. Finite Sample Space S = { w 1 ,...,w N } with p i = p ( { w i } ) = 1 /N, i = 1 ,...,N. For any A ⊂ S we have P ( A ) = | A | | S | = | A | N . 2.1 Counting techniques Consider the set C = { c 1 ,c 2 ,c 3 } and list all possible orderings consisting of two elements of C. With replacement we get c 1 c 1 ,...,c 3 c 3 without replacement we remove c i c i ,i = 1 , 2 , 3 . How many subsets of size two can we form from C = { c 1 ,c 2 ,c 3 } . The subsets are { c 1 ,c 2 } , { c 1 ,c 3 } and { c 2 ,c 3 } . In general we are interest in answering these questions for sets of n elements where we have to find either all the orderings or all of the subsets consisting of r ≤ n elements of C. There are n r orderings if we allow replacements because we have n choices for every position giving rise to n r choices. Here r is not restricted to be smaller than n. If replacements are not allowed then r ≤ n and there are a total of P n r = n ! ( n- r )! 1 orderings (also known as permutations) of r objects out of n . The reasoning here is that there are n choices for the first element, n- 1 choices for the second element and n + 1- r choices for the r th element for a total of n ( n- 1) ... ( n + 1- r ) = n ! ( n- r )! . In particular there are n ! ways of ordering n elements. Sometimes we are only interested in the number of subsets (or combinations) of size r from a set of size n ≥ r . Since each such subset can be ordered in r ! ways it follows that there are n r ¶ = n !...
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Lecture 03 - 1 Lecture Plan • Finite Sample Space(Review...

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