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# Formulas - Useful Formulas and Definitions Summary...

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Useful Formulas and Definitions Summary Statistics 1. Quantile/Percentile of Observation i = (Number of Obs i )/(Total Number of Obs) 2. Median: The smallest number A such that percentile( A ) 0 . 5. 3. Lower Quartile: The 25 th quantile. 4. Upper Quartile: The 75 th quantile. 5. The Mode: The most frequent value in the sample. 6. The Mean: ¯ X = 1 n n i =1 X i 7. Range: (Largest Observation) - (Smallest Observation) 8. Interquartile Range (IQR) = (Upper Quartile) - (Lower Quartile) 9. Mean Absolute Deviation (MAD): 1 n n i =1 | X i - ¯ X | 10. Mean Squared Deviation (MSD): 1 n n i =1 ( X i - ¯ X ) 2 11. Sample Variance: s 2 X = 1 n - 1 n i =1 ( X i - ¯ X ) 2 12. Sample Standard Deviation: s X = q 1 n - 1 n i =1 ( X i - ¯ X ) 2 13. Linear transformations: Suppose Y i = a + bX i , then: (a) ¯ Y = 1 n n i =1 Y i = a + b ¯ X (b) s 2 Y = 1 n - 1 n i =1 ( Y i - ¯ Y ) 2 = b 2 s 2 X Random Variables 1. For Discrete Random Variables: (a) Probability Mass Function (pmf): p ( a ) = P ( X = a ). (b) Cumulative Distribution Function (cdf): F ( a ) = P ( X a ). (c) Mean: μ = xp ( x ) (d) Variance: σ 2 = ( x - μ ) 2 p ( x ) (e) Standard Deviation σ = p ( x - μ ) 2 p ( x ) 2. Binomial distribution: Let n be the number trials, let π be the probability of success at each trial and s the number of successes. Then the pmf of a Binomial Distribution is given by p ( s ) = n ! s !( n - s )! π s (1 - π ) n - s . 3. For Continuous Random Variables: (a) Probability Density Function (pdf) satisfies: P ( a X b ) = R b a p ( x ) dx 1

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(b) Cumulative Distribution Function (cdf): F ( a ) = P ( X a ).
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