# Final Formula Sheet_83016692.pdf - 1 Useful Formulas 1.1...

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1 Useful Formulas 1.1 General Arithmetic Series: n X i =1 i = n ( n + 1) 2 Geometric Series: n X i =0 a i = 1 - a n +1 1 - a , a R X i =0 a i = 1 1 - a , ∀| a | < 1 Binomial Thm: ( x + y ) n = n X i =0 n i x i y n - i x, y R , n N Exponential Fn: X i =0 a i i ! = e a , a R Gamma Fn: Γ( a ) = Z 0 x a - 1 e - x dx, a > 0 Γ( n ) = ( n - 1)! , n N 1.2 Probability Law of Total Probability: partition { A i } n i =1 P ( B ) = n X i =1 P ( B A i ) Bayes Rule: partition { A i } n i =1 P ( A k | B ) = P ( B | A k ) P ( A k ) n i =1 P ( B | A i ) P ( A i ) Multiplication Rule: P n \ i =1 A i ! = P ( A 1 ) P ( A 2 | A 1 ) × · · · × P ( A n | A 1 , . . . , A n - 1 ) Inclusion-Exclusion Principle: P n [ i =1 A i ! = n X i =1 P ( A i ) - X 1 i<j n P ( A i A j )+ + X 1 i<j<k n P ( A i A j A j ) . . . + ( - 1) n - 1 P ( A 1 ∩ · · · ∩ A n ) 1.3 Calculus Change of Variables: Z b a f ( x ) dx ( x = h ( u )) = Z h - 1 ( b ) h - 1 ( a ) f ( h ( u )) h 0 ( u ) du Fundamental Theorem of Calculus: F 0 ( x ) = f ( x ) Z b a f ( x ) dx = F ( b ) - F ( a ) F ( x ) = Z x a f ( u ) du

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