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crib_sheet1 - Crib Sheet for Exam#1 Statistics 211 1...

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Crib Sheet for Exam #1 Statistics 211 1 Chapter 1: Descriptive Statistics Sample Average Population Average ¯ x = 1 n n i =1 x i μ = 1 N N i =1 x i To caluclate the p ’th percentile x [ p ] : 1. Let x ( i ) refer to our data set in ascending order. 2. Let i p = np/ 100. 3. Find the first index i such that i > i p . 4. The p ’th percentile is then: x [ p ] = x ( i - 1) + x ( i ) 2 if i - 1 = i p x ( i ) otherwise s 2 = 1 n - 1 ( x i - ¯ x ) 2 = 1 n - 1 x 2 i - ( ∑ x i ) 2 n Chebychev’s Rule: The proportion of observations that are within k standard deviations ( p k ) of the mean is at least: p k = 1 - 1 k 2 2 Chapter 2: Probability Multiplication rule Permutation Combination n 1 × n 2 × n 2 . . . × n k P k,n = n ! ( n - k )! ( n k ) = P k,n k ! = n ! k !( n - k )! For any two events A and B : P ( A B ) = P ( A ) + P ( B ) - P ( A B ) The conditional probability of A given that B occurred ( P ( B ) > 0): P ( A | B ) = P ( A B ) P ( B ) Two events A and B are independent if P ( A | B ) = P ( A ) If A and B are independent then P ( A B ) = P ( A ) P ( B ) If A, B, C, D, . . . are mutually independent then P ( A B C D . . . ) = P ( A ) P ( B ) P ( C ) P ( D ) . . . 1
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3 Chapter 3: Discrete PDF’s E [ X ] = μ = X S x · p ( x ) E [ h ( X )] = μ h ( x ) = X S h ( X ) · p ( x ) E ( aX + b ) = aE ( X ) + b V ( X ) = σ 2 = E [( x - u ) 2 ] = X S ( x - μ ) 2 · p ( x ) = E [ X 2 ] - E [ X ] 2 V ( aX + b ) = a 2 V ( X ) = a 2 σ 2 3.1 Binomial Distribution For X binomial( n, p ) n = fixed number of trials p = probability of succes ( S ) x = number of successes ( S ) P ( X = x ) = n x p x (1 - p ) n - x x = 0 , 1 , 2 , . . . , n μ = E [ X ] = np σ 2 = V [ X ] = E [( x - μ ) 2 ] = np (1 - p ) 3.2 Multinomial Distribution For X multinomial( n, p 1 , . . . , p r ) n = Number of trials. r = Number of possible outcomes. p i = P (Outcome i on any particular trial).
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