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Unformatted text preview: 5.8 Many Random Variables 187 Figure 5.3 Domain of the Joint Distribution (a) The probability that they meet is given by p = P [ X Y  ≤ 5 ] , which is the probability of being in the crosshatched area of the rectangle. Now, the total area of the rectangle is 60 × 30 = 1800. The area of section A is 10 × 30 = 300, which is also the area of section D. The area of section B is 30 × 30 / 2 = 450, which is also the area of section C. Thus, the area of the crosshatched section is 1800 2 ( 450 + 300 ) = 300. This means that p = 300 / 1800 = 1 / 6 (b) The probability that Ann arrives before Hans is P [ Y < X ] , which is the prob ability of being in the portion of the rectangle above the line Y = X . From the symmetry of the diagram, this can be seen to be equal to 1 / 2. trianglesolid 5.8 Many Random Variables In the previous sections we considered a system of two random variables. In this section we extend the concepts developed for two random variables to systems of more than two random variables. Let X 1 , X 2 ,..., X n be a set of random variables that are defined on the same sample space. Their joint CDF is defined as F X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) = P [ X 1 ≤ x 1 , X 2 ≤ x 2 ,..., X n ≤ x n ] If all the random variables are discrete random variables, their joint PMF is de fined by p X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) = P [ X 1 = x 1 , X 2 = x 2 ,..., X n = x n ] 188 Chapter 5 Multiple Random Variables The properties of the joint PMF include the following: 1. 0 ≤ p X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) ≤ 1 2. summationdisplay x 1 summationdisplay x 2 ··· summationdisplay x n p X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) = 1 3. The marginal PMFs are obtained by summing the joint PMF over the appro priate ranges. For example, the marginal PMF of X n is given by p X n ( x n ) = summationdisplay x 1 summationdisplay x 2 ··· summationdisplay x n 1 p X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) 4. The conditional PMFs are similarly obtained. For example, p X n  X 1 ... X n 1 ( x n  x 1 , x 2 ,..., x n 1 ) = P [ X n = x n  X 1 = x 1 , X 2 = x 2 ,..., X n 1 = x n 1 ] = p X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) p X 1 X 2 ... X n 1 ( x 1 , x 2 ,..., x n 1 ) The random variables are defined to be mutually independent if p X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) = n productdisplay i = 1 p X i ( x i ) If all the random variables are continuous random variables, their joint PDF can be obtained from the joint CDF as follows: f X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) = ∂ n ∂ x 1 ∂ x 2 ...∂ x n F X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) The joint PDF has the following properties: 1. f X 1 X 2 ... X n ( x 1 , x 2 ,..., x n ) ≥ 2....
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 Fall '07
 Carlton
 Probability, Probability theory

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