lecture16

lecture16 - Stat 302 Introduction to Probability Jiahua...

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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 16 January-April 2011 1 / 23 Conditional distribution: continuous random variables Consider the case where X and Y have joint density function f ( x , y ) . Similar to the discrete case, we may attempt to compute P ( Y = y | X = x ) = P ( X = x , Y = y ) P ( X = x ) . Yet this is not feasible because P ( X = x ) = 0. However, the notation of conditional distribution is just as desirable. Jiahua Chen () Lecture 16 January-April 2011 2 / 23 Continuous random variables Suppose f X ( x ) > 0 in a neighborhood of x = a . Let Δ a = [ a , a + δ ] , Δ b = [ b , b + δ ] for some δ > 0. It is seen that P ( X ∈ Δ a ) ≈ f X ( a ) δ > 0. Similarly, P ( X ∈ Δ a , Y ∈ Δ b ) ≈ f ( a , b ) δ 2 . Jiahua Chen () Lecture 16 January-April 2011 3 / 23 Continuous random variables Hence, P ( Y ∈ Δ b | X ∈ Δ a ) δ = f ( a , b ) f X ( a ) which is is well defined. We hence define f Y | X ( y | x ) = f ( x , y ) f X ( x ) as the conditional probability density function of Y given X = x . You may notice that this expression is very close to the expression of the conditional pmf in discrete case. Jiahua Chen () Lecture 16 January-April 2011 4 / 23 Example Suppose the joint pdf of X and Y is given by f ( xy ) = 12 5 x ( 2 − x − y ) for 0 < x < 1 and 0 < y < 1. What is the conditional pdf of X given Y = y ? What is the conditional pdf of Y given X = x ? Jiahua Chen () Lecture 16 January-April 2011 5 / 23 Example Recall the general form of conditional pdf’s f ( x , y ) / f X ( x ) and f ( x , y ) / F Y ( y ) we need only find the marginal pdf’s to answer these two questions. What is the conditional pdf of X given Y = y ? The marginal pdf of Y is f Y ( y ) = integraldisplay 1 x = 12 5 x ( 2 − x − y ) dx = 2 5 ( 4 − 3 y ) . Hence, when x ∈ ( 0, 1 ) , the conditional pdf of y f Y | X ( y | x ) = 6 x ( 2 − x − y ) 4 − 3 y for 0 < y < 1. Jiahua Chen () Lecture 16 January-April 2011 6 / 23 Example Recall the general form of conditional pdf’s f ( x , y ) / f X ( x ) and f ( x , y ) / F Y ( y ) we need only find the marginal pdf’s to answer these two questions....
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