Properties Properties of CDF Ify x thenF X y F X x lim x F X x 0 lim x F X x

Properties properties of cdf ify x thenf x y f x x

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Properties (Properties of CDF) If y x , then F X ( y ) ≥ F X ( x ) . lim x →−∞ F X ( x ) = 0. lim x →∞ F X ( x ) = 1. Definition (Normal/Gaussian random variable) A Normal random variable X with mean μ and variance σ 2 > 0 ( X ∼ N ( μ, σ 2 ) ) has PDF f X ( x ) = 1 2 πσ 2 e 1 2 σ 2 ( x μ ) 2 . We have E [ X ] = μ and Var ( X ) = σ 2 . Remark (Standard Normal) The standard Normal is N ( 0 , 1 ) . Proposition (Linearity of Gaussians) Given X ∼ N ( μ, σ 2 ) , and if a 0, then aX + b ∼ N ( + b, a 2 σ 2 ) . Using this Y = X μ σ is a standard gaussian. Conditioning on an event, and multiple continuous r.v. Definition (Conditional PDF given an event) Given a continuous random variable X and event A with P ( A ) > 0, we define the conditional PDF as the function that satisfies P ( X B A ) = B f X A ( x ) d x. Definition (Conditional PDF given X A ) Given a continuous random variable X and an A R , with P ( A ) > 0: f X X A ( x ) = 1 P ( A ) f X ( x ) , x A, 0 , x /∈ A. Definition (Conditional expectation) Given a continuous random variable X and an event A , with P ( A ) > 0: E [ X A ] = −∞ f X A ( x ) d x. Definition (Memorylessness of the exponential random variable) When we condition an exponential random variable X on the event X > t we have memorylessness, meaning that the “remaining time” X t given that X > t is also geometric with the same parameter i.e., P ( X t > x X > t ) = P ( X > x ) . Theorem (Total probability and expectation theorems) Given a partition of the space into disjoint events A 1 , A 2 , . . . , A n such that i P ( A i ) = 1 we have the following: F X ( x ) = P ( A 1 ) F X A 1 ( x ) + ⋯ + P ( A n ) F X A n ( x ) , f X ( x ) = P ( A 1 ) f X A 1 ( x ) + ⋯ + P ( A n ) f X A n ( x ) , E [ X ] = P ( A 1 ) E [ X A 1 ] + ⋯ + P ( A n ) E [ X A n ] . Definition (Jointly continuous random variables) A pair (collection) of random variables is jointly continuous if there exists a joint PDF f X,Y that describes them, that is, for every set B R n P (( X, Y ) ∈ B ) = B f X,Y ( x, y ) d x d y. Properties (Properties of joint PDFs) f X ( x ) = −∞ f X,Y ( x, y ) d y . F X,Y ( x, y ) = P ( X x, Y y ) = x −∞ [ y −∞ f X,Y ( u, v ) d v ] d u . f X,Y ( x ) = 2 F X,Y ( x,y ) ∂x ∂y . Example (Uniform joint PDF on a set S ) Let S R 2 with area s > 0, then the random variable ( X, Y ) is uniform over S if it has PDF f X,Y ( x, y ) = 1 s , ( x, y ) ∈ S, 0 , ( x, y ) /∈ S. Conditioning on a random variable, independence, Bayes’ rule Definition (Conditional PDF given another random variable) Given jointly continuous random variables X, Y and a value y such that f Y ( y ) > 0, we define the conditional PDF as f X Y ( x y ) = f X,Y ( x, y ) f Y ( y ) . Additionally we define P ( X A Y = y ) A f X Y ( x y ) d x . Proposition (Multiplication rule) Given jointly continuous random variables X, Y , whenever possible we have f X,Y ( x, y ) = f X ( x ) f Y X ( y x ) = f Y ( y ) f X Y ( x y ) . Definition (Conditional expectation) Given jointly continuous random variables X, Y , and y such that f Y ( y ) > 0, we define the conditional expected value as E [ X Y = y ] = −∞ xf X Y ( x y ) d x. Additionally we have E [ g ( X )∣ Y = y ] = −∞ g ( x ) f X Y ( x y ) d x.
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  • Fall '15
  • Probability theory, random variable X

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