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lecture32

# lecture32 - Lecture 32 Discrete Probability Sections 6.1...

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Lecture 32 Discrete Probability Sections 6.1 6.2

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Recap: Linear Recurrence Relations A linear, homogenous relation of degree k with constant coefficients k n k n n n a c a c a c a ... 2 2 1 1
Linear Recurrence Relations The basic approach is to look for a solution of the form a n =r n . k n k n n n a c a c a c a ... 2 2 1 1

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Recap: Linear Recurrence Relations Plugging in a n =r n yields Diving everything by r n-k , we get the characteristic equation k n k n n n r c r c r c r ... 2 2 1 1 0 ... 2 2 1 1 k k k k c r c r c r
Second order equations We first focused on second order equations of the form The characteristic equation is Let r1 and r2 be the two roots of the characteristic equation If r1 r2, then the solution to the recurrence is of the form 2 2 1 1 n n n a c a c a 0 2 1 2 c r c r n n n r r a 2 2 1 1

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What if the roots are not distinct For the recurrence Suppose the characteristic equation Has only one root r 0 Then the solution is of the form 2 2 1 1 n n n a c a c a 0 2 1 2 c r c r n n n nr r a 0 2 0 1

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Find a particular solution Solve the associated homogenous equation

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lecture32 - Lecture 32 Discrete Probability Sections 6.1...

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