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Unformatted text preview: Chapter 11 Random Vectors In this chapter, we expand our exposition of continuous random variables to random vectors. Being versed at dealing with multiple random variables is an essential part of statistics, engineering and science in general. Again, our initial survey of this topic revolves around pairs of continuous random variables. More complex scenarios will be considered in the later parts of this chapter. 11.1 Joint Distribution Functions Let X and Y be two random variables. The joint cumulative distribution function of X and Y is defined by F X,Y ( x,y ) = Pr( X ≤ x,Y ≤ y ) x,y ∈ R . Keeping in mind that X and Y are realvalued functions acting on the out comes of a same experiment, we can also write F X,Y ( x,y ) = Pr ( { ω ∈ Ω  X ( ω ) ≤ x,Y ( ω ) ≤ y } ) . From this characterization, we can identify a few properties of the joint CDF; lim y ↑∞ F X,Y ( x,y ) = lim y ↑∞ Pr ( { ω ∈ Ω  X ( ω ) ≤ x,Y ( ω ) ≤ y } ) = Pr ( { ω ∈ Ω  X ( ω ) ≤ x,Y ( ω ) ∈ R } ) = Pr ( { ω ∈ Ω  X ( ω ) ≤ x } ) = F X ( x ) . 133 134 CHAPTER 11. RANDOM VECTORS Similarly, we can write lim x ↑∞ F X,Y ( x,y ) = F Y ( y ). Taking limits in the other direction, we get lim x ↓∞ F X,Y ( x,y ) = lim y ↓∞ F X,Y ( x,y ) = 0 . When the function F X,Y ( · , · ) is totally differentiable, it is possible to define the joint probability density function of X and Y , f X,Y ( x,y ) = ∂ 2 F X,Y ∂x∂y ( x,y ) = ∂ 2 F X,Y ∂y∂x ( x,y ) x,y ∈ R . (11.1) Hereafter, we refer to a pair of random variables as continuous if the corre sponding joint PDF exists and is defined unambiguously through (11.1). When this is the case, standard calculus asserts that the following equation holds, F X,Y ( x,y ) = x∞ y∞ f X,Y ( ξ, ζ ) dζdξ. From its definition, we note that f X,Y ( · , · ) is a nonnegative function which integrates to one, R 2 f X,Y ( ξ, ζ ) dζdξ = 1 . Furthermore, for any admissible set S ⊂ R 2 , the probability that ( X,Y ) ∈ S can be evaluated through the integral formula Pr(( X,Y ) ∈ S ) = R 2 1 S ( ξ, ζ ) f X,Y ( ξ, ζ ) dζdξ = S f X,Y ( ξ, ζ ) dζdξ. (11.2) In particular, if S is the cartesian product of two intervals, S = ( x,y ) ∈ R 2 a ≤ x ≤ b,c ≤ y ≤ d , then the probability that ( X,Y ) ∈ S reduces to the typical integral form Pr(( X,Y ) ∈ S ) = Pr( a ≤ X ≤ b,c ≤ Y ≤ d ) = b a d c f X,Y ( ξ, ζ ) dζdξ. 11.1. JOINT DISTRIBUTION FUNCTIONS 135 Example 89. Suppose that the random pair ( X,Y ) is uniformly distributed over the unit circle. We can express the joint PDF f X,Y ( · , · ) as f X,Y ( x,y ) = 1 π x 2 + y 2 ≤ 1 otherwise ....
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 Fall '07
 Chamberlain
 Normal Distribution, Probability theory, probability density function, dζ dξ

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