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
Size of the power set of a Fnite 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. DiFerent from k-permutaAons since there is no ordering. ±or one k-combinaAon, k! diFerent k-permutaAons. This number is called “ n choose k Ilya Pollak
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: DefniAon ± 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. Ω R Ilya Pollak
Probability Mass FuncAon (PMF) ±ka “probability law,” “probability distribuAon” PMF of a random variable X is the following funcAon: p X ( x ) = P ( X = x ) Ilya Pollak

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Expected value of X E [ X ] = xp X ( x ) x E [ X ] is also called the mean of X Ilya Pollak
Linearity of expectaAon Ilya Pollak

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15_Exam2_review_Spring_2012 - ECE 302 Spring 2012 Review...

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