week7 - week 7 1 Functions of Random variables In some case...

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Unformatted text preview: week 7 1 Functions of Random variables In some case we would like to find the distribution of Y = h ( X ) when the distribution of X is known. Discrete case E x a m p l e s 1. Let Y = aX + b , a 2. Let ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = = y h x Y x X P y h X P y X h P y Y P y p 1 1 ( ) ( ) ( ) = = = + = = b y a X P y b aX P y Y P 1 2 X Y = ( ) ( ) ( ) ( ) ( ) < = = > = + = = = = = 2 y if y if X P y if y X P y X P y X P y Y P week 7 2 Continuous case Examples 1. Suppose X ~ Uniform(0, 1). Let , then the cdf of Y can be found as follows The density of Y is then given by 2. Let X have the exponential distribution with parameter . Find the density for 3. Suppose X is a random variable with density Check if this is a valid density and find the density of . 2 X Y = 1 1 + = X Y ( ) + = elsewhere x x x f X , 1 1 , 2 1 ( ) ( ) ( ) ( ) ( ) y F y X P y X P y Y P y F X Y = = = = 2 2 X Y = week 7 3 Question Can we formulate a general rule for densities so that we dont have to look...
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week7 - week 7 1 Functions of Random variables In some case...

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