27 Bivariate Normal Distribution

EXAMPLE 5-27 Bivariate Normal Distribution

  At the start of this chapter, the length of different dimensions

  dom variables that are not independent is important in many

  of an injection-molded part was presented as an example of

  applications. As stated at the start of the chapter, if the specifi-

  two random variables. Each length might be modeled by a nor-

  cations for X and Y are 2.95 to 3.05 and 7.60 to 7.80 millime-

  mal distribution. However, because the measurements are

  ters, respectively, we might be interested in the probability that

  from the same part, the random variables are typically not

  a part satisfies both specifications; that is, P(2.95 X

  independent. A probability distribution for two normal ran-

  3.05, 7.60 Y 7.80).

  Bivariate Normal

  Probability

  The probability density function of a bivariate normal distribution is

  Density Function

  1 1 1x

  X 2

  f XY 1x, y; X , Y , X , Y , 2 2 exp e

  2 2 2 2 X Y 21 11 2c X

  2 1x

  X 21 y

  Y 2 1y Y 2 (5-20) X Y 2 df Y

  for

  x

  and

  y

  , with parameters X 0, Y 0,

  X ,

  Y , and 1 1.

  JWCL232_c05_152-190.qxd 1710 2:33 PM Page 178

  CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

  Figure 5-17 Examples of bivariate normal distributions.

  The result that f XY (x, y; X , Y , X , Y , ) integrates to 1 is left as an exercise. Also, the bivari-

  ate normal probability density function is positive over the entire plane of real numbers.

  Two examples of bivariate normal distributions are illustrated in Fig. 5-17 along with corresponding contour plots. Each curve on the contour plots is a set of points for which the probability density function is constant. As seen in the contour plots, the bivariate normal probability density function is constant on ellipses in the (x, y) plane. (We can consider a circle

  to be a special case of an ellipse.) The center of each ellipse is at the point ( X , Y ). If

  0), the major axis of each ellipse has positive (negative) slope, respectively. If 0, the major axis of the ellipse is aligned with either the x or y coordinate axis.

  EXAMPLE 5-28

  The joint probability density function f XY 1x, y2 This probability density function is illustrated in Fig. 5-18.

  1 0.5 1x 2 y 2 Notice that the contour plot consists of concentric circles

  e 12 is a special case of a bivariate normal about the origin.

  distribution with X 1, Y 1, X 0, Y

  0, and 0.

  The following results can be shown for a bivariate normal distribution. The details are left as an exercise.

  Marginal Distributions of

  If X and Y have a bivariate normal distribution with joint probability density f XY (x, y;

  Bivariate Normal

  X , Y , X , Y , ), the marginal probability distributions of X and Y are normal

  Random Variables

  with means X and Y and standard deviations X and Y , respectively. (5-21)

  Figure 5-19 illustrates that the marginal probability distributions of X and Y are normal.

  Figure 5-18 Bivariate normal probability density

  Figure 5-19 Marginal probability

  function with X 1, Y

  1, 0, X 0, and

  density functions of a bivariate Y 0. normal distribution.

  JWCL232_c05_152-190.qxd 1710 2:33 PM Page 179

  5-3 COMMON JOINT DISTRIBUTIONS

  Conditional Distribution of

  If X and Y have a bivariate normal distribution with joint probability density

  Bivariate Normal

  f XY 1x, y; X , Y , X , Y ,

  the conditional probability distribution of Y given

  Random Variables

  X x is normal with mean

  Y

  Y

  Y 冟x

  Y

  X X X x

  and variance

  Y 0 x Y 11

  Furthermore, as the notation suggests, represents the correlation between X and Y. The following result is left as an exercise.

  Correlation of Bivariate Normal

  If X and Y have a bivariate normal distribution with joint probability density function

  Random Variables

  f XY (x, y; X , Y , X , Y , ), the correlation between X and Y is .

  (5-22)

  The contour plots in Fig. 5-17 illustrate that as moves from zero (left graph) to 0.9 (right graph), the ellipses narrow around the major axis. The probability is more concentrated about

  a line in the (x, y) plane and graphically displays greater correlation between the variables. If

  1 or 1, all the probability is concentrated on a line in the (x, y) plane. That is, the probability that X and Y assume a value that is not on the line is zero. In this case, the bivari- ate normal probability density is not defined.

  In general, zero correlation does not imply independence. But in the special case that X and Y have a bivariate normal distribution, if

  0, X and Y are independent. The details are

  left as an exercise.

  For Bivariate Normal Random

  If X and Y have a bivariate normal distribution with

  0, X and Y are independent.

  Variables Zero

  (5-23)

  Correlation Implies

  Independence

  An important use of the bivariate normal distribution is to calculate probabilities involving two correlated normal random variables.