15_Exam2_review_Spring_2012

15_Exam2_review_Spring_2012 - ECE 302 Spring 2012 Review...

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ECE 302 Spring 2012 Review for Exam 2 Ilya Pollak Purdue University
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PermutaAons PermutaAons: number of ways of ordering n objects = n ·∙ (n-­‐1) ·∙ (n-­‐2) ·∙ … ·∙ 2 ·∙ 1 = n! (called n factorial ”). k-­‐permutaAons: number of disAnct k-­‐object sequences out of n objects = n ·∙ (n-­‐1) ·∙ … ·∙ (n-­‐k+1) = n!/(n-­‐k)! Ilya Pollak
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Size of the power set of a finite set Power set of A is the set of all subsets of A. What is the number of subsets of {1,2,…,n}? Each element can either be included in a subset or not: 2 ·∙ 2 ·∙ 2 ·∙ … ·∙ 2 = 2 n . n times Ilya Pollak
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CombinaAons CombinaAons: k-­‐element subsets of a given n-­‐element set. Different from k-­‐permutaAons since there is no ordering. For one k-­‐combinaAon, k! different k-­‐permutaAons. This number is called “ n choose k Ilya Pollak
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ParAAons How many ways are there to divide n items into r groups so that the i -­‐th group has n i items (where n 1 + n 2 + … + n r = n )? Example: if r = 2, the answer is Ilya Pollak Answer: n ! n 1 ! n 2 ! n r ! n n 1 , n 2 , , n r = multinomial coefficients . The reason for this name is that x 1 + x 2 + + x r ( ) n = n n 1 , n 2 , , n r n 1 + n 2 + + n r = n x 1 n 1 x 2 n 2 x r n r
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Random Variables: DefiniAon A random variable is an assignment of a value (number) to every outcome in the sample space. This number is called the numerical value or the experimental outcome of the random variable. In other words, a random variable is a funcAon from the sample space to the set of real numbers.
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