NotesOct27 - The product of independent random variables...

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The product of independent random variables Let X 1 and X 2 be independent continuous ran- dom variables with densities f X 1 and f X 2 ac- cordingly. Let Z = XY . The density of Z is f Z ( z ) = Z -∞ 1 | v | f X ( z/v ) f Y ( v ) dv.
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Example : Let X and Y be iid standard uniform random variables. Find the pdf of the product Z = XY .
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Conditional distributions Let ( X, Y ) be a random vector. A marginal distribution vs. a conditional dis- tribution : the marginal distributions of X consists of probabilities associated with X separately, without regard to Y . if the value y of Y is known, then the con- ditional distribution of X given Y = y consists of probabilities associated with X given that value of Y .
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Conditional distributions in the discrete case Suppose that ( X, Y ) is discrete with a joint pmf p X,Y ( x i , y j ). Then P ( X = x i | Y = y j ) = p X,Y ( x i , y j ) p Y ( y j ) , ( p Y is the marginal pmf of Y ). This is the conditional pmf of X given Y = y j . Notation : p X | Y ( x i | y j ).
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The conditional pmf of X given Y = y j is p X | Y ( x i | y j ) = p X,Y ( x i , y j ) p Y ( y
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