How would you prove this COS 424SML 302 Probability and Statistics Review

# How would you prove this cos 424sml 302 probability

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How would you prove this? COS 424/SML 302 Probability and Statistics Review February 6, 2019 30 / 69

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Continuous random variables Random variables can be continuous. We need a density p ( x ), which integrates to one. If x ∈ < then Z -∞ p ( x ) dx = 1 Probabilities are integrals over smaller intervals. E.g., P ( X ( - 2 . 4 , 6 . 5)) = Z 6 . 5 - 2 . 4 p ( x ) dx COS 424/SML 302 Probability and Statistics Review February 6, 2019 31 / 69
Example: Gaussian distribution Continuous distribution: Gaussian The Gaussian (or Normal) distribution is a continuous distribution, meaning that its support is on continuous numbers. p ( x | μ, σ ) = 1 2 πσ exp - ( x - μ ) 2 2 σ 2 The density of a point x is proportional to the negative exponentiated half distance to μ scaled by σ 2 . Parameters: μ is called the mean ; σ 2 is called the variance . COS 424/SML 302 Probability and Statistics Review February 6, 2019 32 / 69

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Notation for random variables Discrete RVs p denotes the probability mass function , which is the same as the probability of atoms P . We can use P and p interchangeably for discrete distributions. Continuous RVs p is the probability density function the probability of any single value is always zero: P ( X = x ) = 0. We cannot use P and p interchangeably for continuous distributions. This is an unpleasant detail, but mathematically important. COS 424/SML 302 Probability and Statistics Review February 6, 2019 33 / 69
Expectation Consider a function f ( · ) of a discrete random variable X . (Note: f ( X ) is also a random variable.) The expectation is a weighted average of f ( · ), where the weight of each atom is p ( x ), E [ f ( X )] = X x Ω p ( x ) f ( x ) In the continuous case, the expectation is an integral E [ f ( X )] = Z Ω p ( x ) f ( x ) dx COS 424/SML 302 Probability and Statistics Review February 6, 2019 34 / 69

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Conditional expectation The conditional expectation is defined similarly E [ f ( X ) | Y = y ] = X x Ω p ( x | y ) f ( x ) Examples: conditional expectations Given someone’s height Y , what is their expected shoe size X ? Given someone’s zip code, what is the expected number of dollars they will spend on your website? E [ f ( X ) | Y ] is a (function of) random variable Y . COS 424/SML 302 Probability and Statistics Review February 6, 2019 35 / 69
Probability models Probability distributions are simple models of data that we observe. Assume that data are generated from a specific distribution. Infer the parameters of that distribution from the observed data. Interpret those properties in terms of properties of the underlying observed data. COS 424/SML 302 Probability and Statistics Review February 6, 2019 36 / 69

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Probability models, examples Examples of inferences we can make about specific data sets using a simple distribution: Inference Observation the bias of a coin 1000 coin flips the average height of a student 190 students the chance that a politician will win a primary 1200 people in Iowa the proportion of gold in a mountain 20 cubic meters of the mountain the number of bacteria species in our body 4 samples of the gut the evolutionary rate at which DNA mutates 34 completely sequenced mammals What distribution should be used in order to make each inference?
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