10 Independent Random Variables An orthopedic physician’s practice considers the number

Example 5-10 Independent Random Variables An orthopedic physician’s practice considers the number

  of errors in a bill and the number of X-rays listed on the bill. There may or may not be a relation-

  ship between these random variables. Let the random variables X and Y denote the number of errors and the number of X-rays on a bill, respectively.

  Assume that the joint probability distribution of X and Y is defi ned by f XY ( x, y ) in Fig. 5-10(a). The marginal probability distributions of X and Y are also shown in Fig. 5-10(a). Note that

  f XY ( x, y ) = fxfy X () Y .

  The conditional probability mass function f Yx | () y is shown in Fig. 5-10(b). Notice that for any x, f Yx ⏐ () y = fy Y . That is, knowledge of whether or not the part meets color specifi cations does not change the probability that it meets length specifi cations.

  By analogy with independent events, we defi ne two random variables to be independent

  whenever f XY ( x, y ) = fxfy X () Y for all x and y. Notice that independence implies that

  f XY ( x, y ) = fxfy X () Y for all x and y. If we fi nd one pair of x and y in which the equality fails,

  X and Y are not independent. If two random variables are independent, then for f x X ( ) > 0,

  f XY () x, y f xf Y y

  f Yx | () y 5 () f Y y

  f X x

  f X x

  With similar calculations, the following equivalent statements can be shown.

  Independence

  For random variables X and Y , if any one of the following properties is true, the others are also true, and X and Y are independent .

  (1) f XY ()() x, y 5 f X xf Y () y for all and x y (2) f Yx | () y 5 f Y y for all x and y with f X x . 0 (3) f Xy u () x 5 f X x for all and y with f Y () y . 0 (4) PX ( ∈ ∈ A, Y B ) 5 PX ( ∈ APY ) ( ∈ B ) for any sets

  A and B in the range of X and Y , respectively.

  (5-7)

  f Y (y) 4 f Y (y) 4

  FIGURE 5-10

  (a) Joint and mar- ginal probability

  distributions of X and Y for Example

  5-10. (b) Conditional probability

  distribution of Y

  given X x = for

  f X (x)

  0.7 0.2 0.05 f X (x) 0.7 0.2 0.05

  Example 5-10.

  (a)

  (b)

  Chapter 5Joint Probability Distributions

  Rectangular Range for (X, Y ) Let D denote the set of points in two-dimensional space that receive positive probability under

  f XY ( x, y ). If D is not rectangular, X and Y are not independent because knowledge of X can restrict the range of values of Y that receive positive probability. If D is rectangular, independ- ence is possible but not demonstrated. One of the conditions in Equation 5-6 must still be verifi ed.

  The variables in Example 5-2 are not independent. This can be quickly determined because the range of (X, Y ) shown in Fig. 5-4 is not rectangular. Consequently, knowledge of X changes the interval of values for Y with nonzero probability.