15_Exam2_review_Spring_2012

# Random variables condiaonal pmf pxy of x given y is

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 |...
View Full Document

## This note was uploaded on 05/28/2012 for the course ECE 302 taught by Professor Gelfand during the Spring '08 term at Purdue.

Ask a homework question - tutors are online