Theorem 3 if e x 2 and e y 2 then e xy 2 e x 2 e y 2

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Theorem 3 If E [ X 2 ] < and E [ Y 2 ] < then | E [ XY ] | 2 E [ X 2 ] E [ Y 2 ] . For example, | cov( X, Y ) | 2 var( X ) var( Y ) implying | ρ | ≤ 1 . Observe that var( X ) = E [ X 2 ] - E [ X ] 2 0 using Schwartz and Jensen inequalities. Conditional Expectations and Distributions Suppose X is a discrete r.v. Now we define the conditional distribution of another r.v. Y given X = x k (at some point where P ( X = x k ) > 0 ) by F Y | X ( y | x k ) = P ( Y y | X = x k ) = P ( Y y, X = x k ) P ( X = x k ) . By the law of total probability, F Y ( y ) = X k F Y | X ( y | x k ) p X ( x k ) . As we discussed before, when we condition on an event, we are shrinking the sample space under consideration. So there is some normalization that takes place. We also define E [ Y | X = x k ] = Z -∞ ydF Y | X ( y | x k ) . Note that this depends on the value of x k ; it is a function of x k . Let us now take the expectation with respect to X : E X [ E [ Y | X = x k ]] = X k E [ Y | X = x k ] P X ( x k ) = E [ Y ] . We can think of E [ Y | X = x k ] as a discrete random variable that is a function of X . For a discrete r.v. X , the function F Y | X ( y | x k ) could be either a discrete or a continuous r.v. Discrete: p Y | X ( y | x k ) = P ( Y = y | X = x k ) Continuous: There exists a function f Y | X such that F Y | X ( y | x k ) = Z y -∞ f Y | X ( z | x k ) dz. We can also write F Y | X ( y | x k ) = E [ I ( -∞ ,y ] ( Y ) | X = x k ] If Y is discrete we have p Y | X ( y | x k ) = P ( Y = y, X = x k ) p x ( x k ) = p xy ( x k , y ) p x ( x k ) .
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ECE 6010: Lecture 2 – More on Random Variables 14 When X is a continuous r.v., conditional probabilities and expectations are somewhat more complicated, because P ( X = xk ) = 0 for any particular value of x . Recall that E [ Y | X k ] = g ( x k ) for some function g , and E [ g ( x )] = E [ Y ] . Definition 14 Suppose Y is an r.v. on the probability space , F , P ) with E [ | Y | ] < . Then for A ∈ F , define Z A Y dP = E [ I A ( Y )] . 2 Definition 15 Suppose X and Y are random variables and E [ | Y | ] < . The conditional expectation of Y given X = x is any measurable function g ( x ) = E [ Y | X = x ] of x satisfying Z B E [ Y | X = x ] P X ( dx ) = Z X - 1 ( B ) Y dP (2) for all B ∈ B , where X - 1 ( B ) = { ω Ω : X ( ω ) B } . 2 1. It can be shown that under the stated conditions, such a function always exists. 2. If X is discrete then E [ Y | X = x k ] as defined earlier satisfies the property. 3. E [ Y | X = x ] is unique, in the sense that if there are two functions g ( x ) and h ( x ) both satisfying (2) then P ( g ( x ) = h ( x )) = 1 . When a condition is true with probability 1, we say that it is true “almost surely,” or “a.s.” Once we have defined conditional expectation, we can define a conditional c.d.f.: F Y | X ( y | x ) = E [ I ( ,y ] ( y ) | X = x ] . Properties: 1. This definition agrees with the previous one when X is discrete. 2. F Y ( y ) = R R F Y | X ( y | x ) P X ( dx ) . 3. F Y | X is a c.d.f. as a function of y because it satisfies all the properties of a c.d.f. 4. If X and Y are jointly continuous then F Y | X ( y | x ) has a density for every x , f Y | X ( y | x ) = f XY ( x, y ) f X ( x ) There is another interpretation: F Y | X ( y | x ) = lim Δ x 0 + P ( Y y | x - Δ x/ 2 < X x + Δ x/ 2) = lim Δ x 0 + P ( Y y, x - Δ / 2 < X x + Δ x/ 2) / Δ x lim Δ x 0 + P ( x - Δ / 2 < X x + Δ x/ 2) / Δ x = ∂x F XY ( x, y ) ∂x F X ( x ) = ∂x F XY ( x, y ) f x ( x )
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ECE 6010: Lecture 2 – More on Random Variables 15 If X and Y are jointly continuous then ∂y F Y | X ( y | x )
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  • Fall '08
  • Stites,M
  • Probability theory, lim

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