15_Exam2_review_Spring_2012

Random variables condiaonal pmf pxy of x given y is

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Unformatted text preview: summing over all the x j ' s other than x i , for example, pX 1 ( x1 ) = ∑ ∑… ∑ pX 1 ,X 2 ,…,X n ( x1, x 2 ,…, x n ) x2 x3 xn Ilya Pollak FuncAons of many discrete random variables Let Y = g( X1, X 2 ,…, X n ). Then pY ( y ) = ∑p X 1 , X 2 ,…, X n {( x1 ,…, x n )| g ( x1 ,…, x n )= y } E [ g( X1, X 2 ,…, X n )] = ( x1, x 2 ,…, x n ) ∑ g( x ,…, x 1 n ) pX 1 ,…,X n ( x1,…, x n ) x1 ,…, x n If g( X1, X 2 ,…, X n ) = a0 + a1 X1 + a2 X 2 + … + an X n , then E [ g( X1, X 2 ,…, X n )] = a0 + a1 E [ X1 ] + a2 E [ X 2 ] + … + an E [ X n ] Ilya Pollak CondiAoning a random variable on an event •  CondiAonal PMF of a random variable X, condiAoned on an event A with P(A)>0: p X | A ( x ) = P( X = x | A) = P({ X = x} ∩ A) P( A) Ilya Pollak CondiAoning a random variable on another random variable •  If X and Y are random variables, condiAonal PMF pX|Y of X given Y is defined as follows, for all y such that pY(y) >0: pX |Y ( x | y ) = P( X = x | Y = y ) = P({ X = x} ∩ {Y = y}) P({Y = y}) pX ,Y ( x, y ) = pY ( y ) Ilya Pollak CondiAonal means and variances E [ X | Y = y ] = ∑ xp X |Y ( x | y) x Ilya Pollak CondiAonal means and variances p X ,Y ( x, y ) 1 E [ X | Y = y ] = ∑ xp X |Y ( x | y) = ∑ x = ∑ xpX ,Y ( x, y) pY ( y) pY ( y) x x x Ilya Pollak CondiAonal means and variances p X ,Y ( x, y ) 1 E [ X | Y = y ] = ∑ xp X |Y ( x | y) = ∑ x = ∑ xpX ,Y ( x, y) pY ( y) pY ( y) x x x 2 var( X | Y = y) = E ⎡( X − E [ X | Y = y ]) | Y = y ⎤ ⎣ ⎦ Ilya Pollak CondiAonal means and variances p X ,Y ( x, y ) 1 E [ X |...
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This note was uploaded on 05/28/2012 for the course ECE 302 taught by Professor Gelfand during the Spring '08 term at Purdue.

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