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Basic Statistics Cheat Sheet Term Description Probability The proportion of units with X = x in the population P ≜ 𝑃(𝑋 = 𝑥) Joint Probability The proportion of units with X=x and Y=y in the population P ≜ 𝑃(𝑋 = 𝑥, 𝑌 = 𝑦) Conditional Probability The proporiton of units with X = x in the subset of the populaton with Y=y ≜ 𝑃(𝑋 = 𝑥|𝑌 = 𝑦) = 𝑃(𝑋 = 𝑥, 𝑌 = 𝑦)/𝑃(𝑌 = 𝑦) Probability Distribution The distribution of 𝑃(𝑋 = 𝑥) for all possible values x. The sum over all possible values is 1. For categorical variables: ∑𝑃(𝑋 = 𝑥) = 1 𝑥 For continuous variables ∫ 𝑓(𝑥)𝑑𝑓 ∞ −∞ = 1 Where 𝑓(𝑥) is known as the probability density function Independence If X and Y are independent (𝑌 ∐ 𝑋)then 𝑃(𝑌 = 𝑦|𝑋 = 𝑥) = 𝑃(𝑌 = 𝑦) and 𝑃(𝑋 = 𝑥|𝑌 = 𝑦) = 𝑃(𝑋 = 𝑥) for every value x and y Conditional Independence If X and Y are conditionally independent given Z=z (𝑌 ∐𝑋|𝑍 = 𝑧)then 𝑃(𝑌 = 𝑦|𝑋 = 𝑥, 𝑍 = 𝑧) = 𝑃(𝑌 = 𝑦|𝑍 = 𝑧) And 𝑃(𝑋 = 𝑥|𝑌 = 𝑦,𝑍 = 𝑧) = 𝑃(𝑋 = 𝑥|𝑍 = 𝑧) For every value x and y Bayes Rule 𝑃(𝑌 = 𝑦|𝑋 = 𝑥) = 𝑃(𝑋 = 𝑥|𝑌 = 𝑦)𝑃(𝑌 = 𝑦) 𝑃(𝑋 = 𝑥) Law of total probability 𝑃(𝑌 = 𝑦) = ∑ 𝑃(𝑌 = 𝑦,𝑋 = 𝑥) 𝑥 = ∑𝑃(𝑌 = 𝑦|𝑋 = 𝑥)𝑃(𝑋 = 𝑥) 𝑥 Expected Value (mean) 𝐸[𝑋] = 𝜇𝑋 = ∑𝑥𝑃(𝑋 = 𝑥) 𝑥 𝐸[𝑔(𝑋)] = 𝜇𝑔(𝑋) = ∑𝑔(𝑥)𝑃(𝑋 = 𝑥) 𝑥 𝐸[𝑌|𝑋 = 𝑥] = 𝜇𝑌|𝑋 = ∑𝑦𝑃(𝑌 = 𝑦|𝑋 = 𝑥) 𝑦 Term Description Variance Standard Deviation 𝑉𝑎𝑟(𝑋) = 𝜎𝑥2 = 𝐸[((𝑋 −𝜇𝑋)2)] = ∑(𝑥 −𝜇𝑋)2 𝑃(𝑋 = 𝑥) 𝑥 𝑆𝑡𝑑𝐷𝑒𝑣(𝑋) = √𝑉𝑎𝑟(𝑋) = √𝜎𝑥2 = 𝜎𝑥 A measure of how "spread out" the data are. 𝑉𝑎𝑟(𝑎𝑋 +𝑏) = 𝑎2𝑉𝑎𝑟(𝑋) Covariance 𝐶𝑜𝑣(𝑋,𝑌) = 𝜎𝑋𝑌 ≜ 𝐸[(𝑋 −𝜇𝑋)(𝑌 −𝜇𝑌)] = ∑(𝑥 −𝜇𝑋)(𝑦−𝜇𝑌)𝑃(𝑋 = 𝑥,𝑌 = 𝑦) 𝑥,𝑦 The degree to which two variables are "linearly associated". 𝐶𝑜𝑣(𝑎𝑋 +𝑏𝑌, 𝑍) = 𝑎𝐶𝑜𝑣(𝑋,𝑍)+𝑏𝐶𝑜𝑣(𝑌,𝑍) Variance of sum of 2 random variables 𝑉𝑎𝑟(𝑋 +𝑌) = 𝑉𝑎𝑟(𝑋)+𝑉𝑎𝑟(𝑌)+2𝐶𝑜𝑣(𝑋,𝑌) Correlation Coefficient 𝜌𝑋𝑌 = 𝜎𝑋𝑌 𝜎𝑋 𝜎𝑌 Normalized covariance, dimensionless ranging from -1 to 1 Regression Coefficient Simple regression: Y = a + b X b = RYX = σXY σX2 Multiple regression: 𝑌 = 𝑎 + 𝑏 𝑋 +𝑐𝑍 𝑏 = 𝑅𝑌𝑋|𝑍 = 𝜎𝑍2 𝜎𝑋𝑌 −𝜎𝑍𝑌𝜎𝑋𝑍 𝜎𝑋2 𝜎𝑍2 −𝜎𝑋𝑍2