15 Probability as a Ratio of Volumes

EXAMPLE 5-15 Probability as a Ratio of Volumes

  Suppose the joint probability density function of the con-

  The region that receives positive probability is a circle of

  tinuous random variables X and Y is constant over the re-

  radius 2. Therefore, the area of this region is 4

  . The area of

  y

  gion x 2 2

  Determine the probability that

  the region

  x 2

  y

  1 is . Consequently, the requested

  X 2

  Y

  1. probability is 14 2 1 4.

  Marginal Probability

  If the joint probability density function of continuous random variables X 1 , X 2 ,p,X p

  Density

  is the f X 1 X 2 p X p

  1x 1 ,x 2 ,p,x p 2 , marginal probability density function of is X i

  Function

  冮冮 冮

  f X i 1x i 2 p f X 1 X 2 p X p 1x 1 ,x 2 ,p,x p 2 dx 1 dx 2 p dx i 1 dx i 1 p dx p (5-9)

  where the integral is over all points in the range of X 1 ,X 2 ,p,X p for which X i x i .

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  5-1 TWO OR MORE RANDOM VARIABLES

  As for two random variables, a probability involving only one random variable, say, for example, P 1a X i

  b 2, can be determined from the marginal probability distribution of X i

  or from the joint probability distribution of X 1 ,X 2 ,p,X p . That is,

  P 1a X i b 2 P1 X 1 ,p, X i 1 ,a X i b,

  X i 1 ,p, X p 2

  Furthermore, E 1X i 2 and V 1X i 2, for

  1, 2, p , p, can be determined from the marginal probability distribution of X i or from the joint probability distribution of X 1 ,X 2 ,p,X p as

  follows.

  Mean and Variance from Joint Distribution

  E 1X i 2 p x i f X 1 X 2 p X p 1x 1 ,x 2 ,p,x p 2 dx 1 dx 2 p dx 冮 p 冮 冮

  and

  (5-10)

  X i 2 f X 1 X 2 p X p 1x 1 ,x 2 ,p,x p 2 dx 1 dx 2 p dx 冮 p 冮 冮

  V 1X i 2 p

  1x i

  EXAMPLE 5-16

  Points that have positive probability in the joint probability

  P 1X 2 1 2 f X 1 X 2 X 3 12, 1, 02 f X 1 X 2 X 3 10, 1, 22

  distribution of three random variables X 1 ,X 2 ,X 3 are shown

  in Fig. 5-11. The range is the nonnegative integers with x 1 f X 1 X 2 X 3

  x 2 x 3 3. The marginal probability distribution of X 2 is

  P 1X 2 X 2 f 1 X 2 X 3 11, 2, 02 f X 1 X 2 X 3 10, 2, 12

  found as follows.

  With several random variables, we might be interested in the probability distribution of some

  subset of the collection of variables. The probability distribution of X 1 ,X 2 ,p,X k , k p can

  be obtained from the joint probability distribution of X 1 ,X 2 ,p,X p as follows.

  Distribution of a Subset of

  If the joint probability density function of continuous random variables X 1 ,X 2 , p ,X p

  Random Variables

  is the f X 1 X 2 p X p 1x 1 ,x 2 ,p,x p 2, probability density function of X 1 ,X 2 , p ,X k ,k p, is

  f X 1 X 2 p X k 1x 1 ,x 2 ,p,x k 2

  冮冮 冮

  p f X 1 X 2 p X p 1x 1 ,x 2 ,p,x p 2 dx k 1 dx k 2 p dx p (5-11)

  where the integral is over all points in the range of X 1 ,X 2 ,p,X p for which

  X 1 x 1 ,X 2 x 2 ,p,X k x k .

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  CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

  x 3 3 x 2 2 3

  Figure 5-11 Joint

  probability distribution 0

  of X 1 ,X 2 , and X 3 .

  1 2 3 x 1